Journal of Applied Nonlinear Dynamics

Volume 7 Issue 4 December 2018

ISSN 2164-6457 (print) ISSN 2164-6473 (online)

Journal of Applied Nonlinear Dynamics

Journal of Applied Nonlinear Dynamics Editors J. A. Tenreiro Machado ISEP-Institute of Engineering of Porto Dept. of Electrical Engineering Rua Dr. Antonio Bernardino de Almeida 431, 4200-072 Porto, Portugal Fax:+ 351 22 8321159 Email: [email protected]

Albert C.J. Luo Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1805 USA Fax: +1 618 650 2555 Email: [email protected]

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Antonio M Lopes

Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics Moscow State University 119991 Moscow, Russia Fax: +7 495 939 0397 Email: [email protected]

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Continued on back materials

Journal of Applied Nonlinear Dynamics Volume 7, Issue 4, December 2018

Editors J. A. Tenreiro Machado Albert Chao-Jun Luo

L&H Scientific Publishing, LLC, USA

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Journal of Applied Nonlinear Dynamics 7(4) (2018) 329-335

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Switching Tracking Control and Synchronization of Four-Scroll Hyperchaotic Systems K. S. Ojo†, A. B. Adeloye, A. O. Busari Department of Physics, University of Lagos, Akoka, Lagos, Nigeria Submission Info Communicated by A.C.J. Luo Received 1 May 2017 Accepted 5 July 2017 Available online 1 January 2019 Keywords Switching controllers Tracking control Synchronization Anti-synchronization Routh-Hurwitz criterion Four-scroll chaotic system

Abstract The paper investigates switching tracking control and synchronization of four-scroll hyperchaotic systems via the Open Plus Closed Loop (OPCL) technique. Based on Routh Hurwitz criterion, controllers which enable the state variables of the system to either stabilize to a chosen position or track desired smooth time dependent functions at different time intervals are designed. Similarly, controllers which enable trajectories of the drive system to either synchronize or anti-synchronize with the trajectories of the response system at different time intervals are designed. Numerical simulations are presents to validate the effectiveness of the proposed tracking control and synchronization technique. The newly proposed switching tracking control and synchronization scheme could be used to improve the design and control of a switch regulator. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction One of the most active research areas in the field of nonlinear science is the study of chaos control and synchronization with its applications to physical, biological, chemical and financial systems [1]. Chaotic behaviour has been found useful in some applications such as secure communications [2] while it is not useful in some applications [3]. As a result of positive and negative effects of chaos, chaos control and synchronization have been effectively developed to either eliminate, or stabilize chaos where its presence is not desirable and make use of chaos where its presence is desirable [4, 5]. So, chaos control deals with the design of some control parameter to stabilize the chaotic orbit or modify the behavoiur of a system to follow a desired trajectory. On the other hand, synchronization deals with forcing two or more systems of different trajectories to achieve identical dynamics asymptotically with time. Since Pecora and Carrol [8] introduced a method of synchronization of two identical systems with different initial conditions, a variety of approaches have been proposed for the synchronization of chaotic systems, including complete synchronization [8], anti-synchronization [9, 10], generalized synchronization [11], projective synchronization [12], function projective synchronization [13], hybrid synchronization [14], hybrid projective synchronization [15] and hybrid function projective synchronization [16]. In † Corresponding

author. Email address: [email protected]; [email protected]

ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2018.12.001

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K. S. Ojo, A. B. Adeloye, A. O. Busari / Journal of Applied Nonlinear Dynamics 7(4) (2018) 329–335

search of effective methods of achieving a stable synchronization, several control and synchronization methods such active control, backstepping technique, sliding mode control, feedback control, OPCL and many more have been extensively investigated as reported in [6, 16] and the reference therein. A lot of works have been reported on different types of synchronization and control via different methods. To the best of our knowledge, only one research reported was on switching synchronization of fractional order system through the active control technique based on bidirectional configuration scheme. The authors of this paper achieved the switching synchronization by switching the control parameter so that either of the system achieved synchronization or both achieved synchronization such that one system synchronized for a particular time interval while the other synchronized for the remaining time interval. This research work on one hand presents switching tracking control four scroll hyperchaotic system. The control technique is such that the system is controlled to track by switching the control function. On the other hand, using the master slave synchronization scheme, either synchronization or anti synchronization is achieved by switching the scaling parameter at different intervals of time It should be noted that the present research work switched the scaling parameter/function at different time intervals and not the control parameter as reported in [18]. Another point is that in our own case either synchronization or antisynchronization is achieved at different intervals while in the case of [18] only synchronization was achieved. Moreover, master slave configuration was used via the OPCL technique while the previous paper used bidirectional configuration via active control technique. Among the nonlinear methods of synchronization, the OPCL control method is outstanding because it has been practically implemented [7]. It is worthy of note that despite the excellent performance of the OPCL control method, it has not been used to stabilize a chaotic system at a chosen position or track a desired trajectory. It has also been used for the design of switching controllers that are capable of achieving tracking control, synchronization and antisynchronization. As a result of practical application of chaos stabilization and tracking in physical systems we report here for the first time switching control for stabilization, tracking synchronization and antisynchronization of a hyperchaotic system via the OPCL Meanwhile, all the controllers that have been so far designed for control and synchronization of chaotic system using various methods are not flexible enough, thus limit the choice of control laws. The existing designed controllers can therefore, only accomplish one task (that is, synchronize or antisynchronize). Hence it is not suitable for multiple task objectives. As a result of the limitation of the existing designed controller, the present research work presents switching tracking control and synchronization which is able to execute multiple task objectives. The switching controllers are able to change from one task to the other within a specified time interval. Therefore, the switching controllers are more flexible and enable one to optimize the choice of control laws most especially where multiple task objectives is required. The switching controllers are more applicable to real life situations than the existing designed controllers. To the best of our knowledge, switching tracking control and synchronization via the OPCL is considered in this paper for the first time. Motivated by the aforementioned reasons, this paper focuses on switching tracking control and synchronization of four scroll hyperchaotic systems. The paper is organized as follows. Section 2 gives Open-Plus-Closed-Loop (OPCL) theory. Section 3 presents switching tracking control of a four scroll hyperchaotic system via OPCL. Section 4 deals with the switching synchronization of four scroll hyperchaotic systems via OPCL. Section 5 concludes the paper.


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2 Open-Plus-Closed-Loop (OPCL) theory In this section, a brief description of OPCL synchronization is presented. Consider a drive nonlinear system x˙ = Ax + f (x), x ∈ ℜn , (1) while the response system is

y˙ = Cy + f (y), y ∈ ℜn .

(2)

After coupling, the response system is given by y˙ = Cy + f (y) +U

(3)

where A, C are system parameters. and U is the coupling defined as U = g˙ − f (g) + (H − J(g))e,

(4)

where J(g) is the Jacobian of f (g), H is an arbitrary constant Hurwitz matrix (n × n) whose all eigenvalues have negative real parts and goal dynamics g = α (t)x and g = (g1 , g2 , g3 , ...gn ) and α = (α1 , α2 , α3 , ...αn ) are arbitrary scaling constants or functions. The error signal of the the coupled systems is defined by e = (y − g) and f (g) can be be written using Taylor series expansion as f (x) = f (g) +

∂ f (g) e+ ··· . ∂g

(5)

Keeping the first order terms in (5) and substituting it into (4) yields the following error dynamics in (6) e˙ = He (6) since H is an Hurwitz matrix and e → 0 as t → ∞ then stable asymptotic synchronization is obtained. 3 Design of controllers for switching tracking control for four scroll hyperchaotic system via OPCL method In this section, detailed derivation of control functions that are capable of stabilizing each variable of four scroll hyperchaotic system to a chosen position or track desired smooth trajectories is presented. To achieve this goal we use the four scroll hyperchaotic system in (7)     x˙1 ax1 − x2 x3  x˙2    x1 x3 − bx2 =  f (x) =  (7)  x˙3   cx1 x2 − dx3 + gx1 x4  x˙4 f x4 − hx2 and



a  x 3 J( f (x)) =   (cx2 + gx4 ) 0

 −x3 −x2 0 −b x1 0  . cx1 −d gx1  −h 0 f

(8)

From (7) f (g) is 

 ag1 − g2 g3   g1 g3 − bg2  f (g) =   cg1 g2 − dg3 + gg1 g4  f g4 − hg2

(9)

332


while from (8) f (g) is 

a  g3 J( f (g)) =   (cg2 + gg4 ) 0

 −g3 −g2 0 −b g1 0   cg1 −d gg1  −h 0 f

(10)

where (g1 , g2 , g3 , g4 )T are the desired trajectories which may be constants or continuous time-dependent functions. The error system between the state variable of the system and the desired trajectories is     y1 − g1 e1  y2 − g2   e2     e= (11)  y3 − g3  =  e3  . y4 − g4 e4 Substituting an appropriate Hurwitz matrix, time derivative of (g1 , g2 , g3 , g4 )T and (9) yields the control functions in (12)    g˙1 − ag1 + g2 g3 −(1 + a)e1 + g3 e2 + g2 e3    g˙2 − g1 g3 + bg2 −g3 e1 + (b − 1)e2 − g1 e3   U =  g˙3 − cg1 g2 + dg3 − gg1 g4  +  −(cg2 + gg4 )e1 − cg1 e2 + (d1 − 1)e3 − gg1 e4 g˙4 − f g4 + hg2 he2 − (1 + f )e4

(11) into (4) 

 . 

(12)

Using the ode 45 run on Matlab, the dynamics of the hyperchaotic system (7) for the parameter values a = 8, b = 40, c = 2, d = 14, f = 0.05, g = 5, h = 0.2 is shown by two dimensional phase space and the trajectory of the state variables in Fig. 1 and 2. The solution of system (7) with the control functions defined in (12) where g1 = g2 = g3 = g4 is shown in Fig 3. The result in Fig. 3 shows that the system variables are controlled to track the desired trajectory. This result shows the effectiveness of the designed controllers. 4 Switching synchronization of four scroll hyperchaotic systems via OPCL In this section, detailed derivation of control functions that are capable of realizing synchronization between two hyperchaotic systems is presented. To achieve this goal the hyperchaotic system (7) is taken as the drive system while (13) is taken the response system     y˙1 ay1 − y2 y3  y˙2    y1 y3 − by2  =  (13)  y˙3   cy1 y2 − dy3 + gy1 y4  . y˙4 f y4 − hy2 The Jacobian of the response system is 

a  y 3 J(y)) =   (cy2 + gy4 ) 0

 −y3 −y2 0 −b y1 0  . cy1 −d gy1  −h 0 f

(14)

 0 0  . 0  −1

(15)

The Hurwitz matrix is chosen as 

−1  0 J(y) =   0 0

0 −1 0 0

0 0 −1 0

100

100

50

50 x2

x1


0

333

0 b

−50

−50 a 200 400 time

600

−100 0

200

30

100

20 x4

x3

−100 0

0 −100

200 400 time

10 0

c −200 0

600

200 400 time

d 600

−10 0

200 400 time

600

Fig. 1 The dynamics of the state variables with the controllers are deactivated for 0 < t ≤ 100 while the dynamics of the sate variable are applied in a step wise for are: g1 = 0 for 100 < t ≤ 200; g1 = 20 for 200 < t ≤ 300; and g1 = 10 + 5sin(0.04π t) for 300 < t ≤ 600.

The choice of the transformation matrix is given as   0 0 α1 (t) 0  0 α2 (t) 0 0  . α =  0 0 α3 (t) 0  0 0 0 α4 (t) To achieve a desired goal, g is chosen as     g1 x1 0 0 α1 (t) 0  g2   o α2 (t) 0   x2 0  =   g3   0 0 α3 (t) 0   x3 g4 x4 0 0 0 α4 (t)

(16)

 α1 (t)x1   α2 (t)x2   =   α3 (t)x3  . α4 (t)x4 



(17)

Using the expression in (4), the control function that achieves stable synchronization is obtained as     −(1 + a)e1 + g3 e2 + g2 e3 α˙ 1 (t)x1 + α1 (t)x˙1 − ag1 + g2 g3     −g3 e1 + (b − 1)e2 − g1 e3 α˙ 2 (t)x2 + α2 (t)x˙2 − g1 g3 + bg2    U =  α˙ 3 (t)x3 + α3 (t)x˙3 − cg1 g2 + dg3 − gg1 g4  +  −(cg2 + gg4 )e1 − cg1 e2 + (d1 − 1)e3 − gg1 e4  . α˙ 4 (t)x4 + α4 (t)x˙4 − f g4 + hg2 he2 − (1 + f )e4 (18) In other to confirm the effectiveness of the designed control function, ode45 fourth order Runge-Kutta algorithm run on Matlab is used with g1 = g2 = g3 = g4 and α1 = α2 = α3 = α4 = α for a particular time

334


100

50 x1

x2

y1

y2

x2, y2

x1, y1

50 0

0

−50 −100

4

6 time

−50

8

100

8

x4

0.5

y3

x4, y4

x3, y3

6 time

1 x3

50 0 −50 −100

4

y4

0 −0.5

4

6 time

8

−1

4

6 time

8

Fig. 2 Time evolution of the state variables of the drive (solid line) and response (dashed line) variables when the controllers have been activated.

range. Solving the drive system (8) and the response system (9) with control functions defined in (18) using the system parameter values and different initial conditions for the drive and response system to ensure hyperchaotic dynamics of the state variables. In the numerical simulations, synchronization of the state variables is achieved for t ≤ 6 when α is chosen as 1, while anti-synchronization of the state variable is achieved for 6 < t ≤ 8 when α is chosen as -1 as depicted in Fig 4. The results show the effectiveness of the designed controllers. 5 Concluding remarks Switching tracking control and synchronization of four-scroll hyperchaotic Systems has been realized in this paper using the OPCL technique. The OPCL technique has not been used for stabilization or tracking of chaotic systems despite its excellent performance in synchronization of chaotic systems. It is also noticed that most of controllers designed for the tracking control and synchronization of chaotic systems can only be used to achieve only one type of tracking control and synchronization behavior in a chosen dimension or all the dimensions of the chaotic systems. This shows that the previous designed for tracking control and synchronization can only achieve a single task. In order to overcome the present challenge, switching tracking control and synchronization is proposed and achieved via analytical and numerical simulations. The switching tracking control and synchronization is implemented using four-scroll hyperchaotic systems. The designed controllers were implemented such that the control laws can switch from one form to the other as the trajectory moved from one chosen time domain to another. The choice of control law implemented at each time domain was based on the


335

type of control we intended to achieve. From the research work, it is clear that the switching tracking control and synchronization implemented in this paper gives better flexibility and optimization in the choice of the control law to be applied at a chosen time interval. Thus, the switching tracking control is efficient, effective and suitable for both single and multiple task situations. This switching tracking control and synchronization will be very useful in real physical and biological systems where different tasks are to be achieved at different times by the same or different organs of the systems. References [1] Strogatz, S.H. (1994), Nonlinear Dynamics and Chaos: with application to physics, chemistry, biology, and Engineering, Addison-Wesley Publishing Company: USA. [2] Blackledge, J.M. (2008), Multi-algorithmic cryptography using deterministic chaos with applications to mobile communications, international society for advanced science and technology, Transactions on Electronics and Signal Processing, 2(1), 23-64. [3] Lu, J. and Chen, S. (2003), Chaotic time series analysis and its application, Wuhan University Press, 4, pp. 570-578. [4] Njah, A.N. and Ojo, K.S. (2010), Backstepping control synchronization of parametrically and externally excited van der Pol Oscillators with application to secure communications, International Journal of Modern Physics B, 24(3), 4581-4593. [5] Ojo, K.S., Njah, A.N., and Olusola, O.I. (2014), Generalized compound synchronization of chaos in different order chaotic Josephson junctions, International Journal of Dynamics and Control, DOI:10.1007/s40435-0140122-5 (in press). [6] Ojo, K.S., Njah, A.N., Olusola, O.I., and Omeike, M.O. (2014), Generalized reduced-order hybrid combination synchronization of three josephson junctions via backstepping technique, Nonlinear Dynamics, 77, 583-595. [7] KamdoumTamba, V., Fotsin, H.B., Kengne, J., KapcheTagne, F., and Talla, P.K. (2015), Coupled inductorsbased chaotic colpitts oscillators, Mathematical modeling and synchronization issues Eur. Phys. J. Plus, 130-137 (18pp). [8] Pecora, L.M. and Carroll, T.L. (1990), Synchronization of chaotic systems, Physical Review Letter, 64, 821824. [9] Ayub, K,n and Priyamvadam T. (2013), Synchronization, anti-synchronization and hybrid- synchronization of a double pendulum under the effect of external forces, International Journal of Computational Engineering Research, (ijceronline.com), 3(1), 166-176. [10] Ojo, K.S., Njah, A.N., and Adebayo, G.A. (2011), Anti-synchronization of identical and non-identical φ 6 van der pol and φ 6 duffing oscillator with both parametric and external excitations via backstepping approach, International journal of modern physics B, 25(14), 1957-1969. [11] Rulkov, N.F., Sushchik, M.M., and Tsimring, L.S. (1995), Generalized synchronization of chaos in directionally coupled chaotic system, Phys. Rev., E51, 980. [12] Sundarapandian, V. and Karthikeyan, R. (2011), Hybrid chaos synchronization of hyperchaotic lorenz and hyperchaotic chen systems by active nonlinear control, International Journal of Electrical and Power Engineering, 5(5), 186-192. [13] Park, J.H. and Kwon, O.M. (2003), A novel criterion for delayed feedback control of time-delay chaotic systems, Chaos, Solitons and Fractals, 17, 709-716. [14] Ojo, K.S., Njah, A.N., Ogunjo, S.T., and Olusola O.I. (2014), Reduced order function projective combination synchronization of three josephson junctions using backstepping technique, Nonlinear Dynamics and Systems Theory, 14(2), 119-133. [15] Roy, P.K., Hens, C., Grosu, I., and Dana S.K. (2011), Engineering generalized synchronization in chaotic oscillators, Chaos, 21(1), 013106 (7pp). [16] Padmanaman, E., Banerjee, R., and Dana S.K. (2012), Targeting and control of synchronization in chaotic oscillators, International Journal of Bifurcation and chaos, 22, 1250177-(12pp). [17] Sun, Z.W. (2013), Function Projective synchronization of two four-scroll hyperchaotic systems with unknown parameters, Cent. Eur., 11(1), 89-95. [18] Radwan, A.G., Moaddy, K., Salama, K.N., Momani, S., and Hashim, I. (2014), Control and switching Synchronization of fractional order chaotic system using active control, Journal of Advanced Research, 5, 125-132.



Hopf Bifurcation and Stability Analysis of a Predator-Prey System with Holling Type IV Functional Response Z. Lajmiri1 , R. Khoshsiar Ghaziani2†, I. Orak1 1 2

Sama technical and vocational training college, Islamic Azad University , Izeh branch, Izeh Iran Department of Applied Mathematics, Shahrekord University, P.O.Box. 115, Shahrekord, Iran Submission Info Communicated by A.C.J. Luo Received 15 June 2017 Accepted 6 July 2017 Available online 1 January 2019 Keywords Limit cycle Hopf bifurcation Fold bifurcation

Abstract In this paper, we investigate the dynamical complexities of a predator-prey model with Holling type IV functional response, which describes interaction between two populations of prey and predator. We perform a bifurcation analysis of this model analytically and numerically. Our bifurcation analysis indicates that the system exhibits numerous types of codimension one and two bifurcations including fold, subcritical and supercritical Hopf, cusp and Bogdanov-Takens. By numerical continuation method, we also compute several curves of equilibria and bifurcations. Further, by numerical simulations we reveal more complex dynamics of the model. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Ecological systems are characterized by the interaction between species and their natural environment. An important type of interaction which effects population dynamics of all species is predation. Thus predator-prey models have been in the focus of ecological science since the early days of this discipline. It has turned out eventually that predator-prey systems can show different dynamical behaviors (steady states, oscillations, chaos) depending on the value of model parameters. One of the important factors which affect the dynamical properties of biological and mathematical models is the functional response. The formulation of a predator-prey model critically depends on the form of the functional response that describes the amount of prey consumed per predator per unit of time, as well as the growth function of prey [1, 2]. That is a functional response of the predator to the prey density in population dynamics refers to the change in the density of prey attached per unit time per predator as the prey density changes. Two species models like Holling type II, III and IV of predator to its prey have been extensively discussed in the literature [1,3–10]. The predator-prey systems with Holling type functional responses have been studied in many works [11–14]. The purpose of this paper is to study the dynamics of prey-predator model with Holling type IV function involving intra-specific competition. We prove that the model has bifurcation that is associated † Corresponding

author. Email address: [email protected]


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with intrinsic growth rate. The stability analysis that we carried out analytically has also been approved numerically. Most importantly we show that the Hopf bifurcation plays, for various reasons, a crucial role. In this paper we rely heavily on advanced numerical methods, namely numerical continuation to obtain results that cannot be obtained analytically. Numerical bifurcation analysis techniques are very powerful and efficient in physics, biology, engineering, and economics. In this paper we consider the following classical prey- predator system:  dx   = xq(x) − α yp(x), dt (1)   dy = yp(x) − β y, dt where x(0) > 0, y(0) > 0, x, y represent the prey and predator density, respectively. p(x) and q(x) are so-called predator and prey functional response respectively. α , β are the conversion and predator‘s mx refers to as Michaelis-Menten function or death rates, respectively. The functional response p(x) = a+x a Holling type II function, where m > 0 denotes the maximal growth rate of the species, a > 0 is halfmx2 saturation constant. Another class of response function is Holling type III, i.e., p(x) = a+x . In general the response function p(x) satisfies the general hypothesis: p(x) is continuously differentiable function defined on [0,∞] and satisfies p(0) = 0, p(x) > 0, limx→∞ p(x) = m < ∞. The simplified Monod-Haldane or mx Holling type-IV functional is a modification of the Holling type III function [15], in which p(x) = a+x 2. The paper is organized as follows: In Section 2, we introduce the model and discuss the stability and bifurcations of its equilibrium points. In Section 3, we numerically compute curves of codim-1 bifurcations and their corresponding critical normal form coefficients of codim-2 bifurcation points. In Section 4, we summarize our results. 2 Equilibria of the system and their stability The prey-predator systems have been discussed widely in many decades. In literature many studies considered the prey-predator with functional responses. However, considerable evidence that some prey or predator species have functional response because of the environmental factors. It is more appropriate to add the functional responses to these models in such circumstances. For instance, a system suggested in model (1), where x(t) and y(t) represent densities or biomasses of the prey species and predator species respectively; p(x) and q(x) are the intrinsic growth rates of the predator and prey mx respectively; α , β , are the death rates of prey and predator respectively. If p(x) = 1+x 2 , q(x) = ax(1 − x), and a = 1, that is where a is the half-saturation constant in the Holling type IV functional response, then model (1) becomes:  dx α my  = x[a(1 − x) − ],  dt 1 + x2 (2)   dy = y[ mx − β ], dt 1 + x2 where a, α , β , m are all positive parameters. We now introducing intra-specific competition, the (2) becomes:  dx α my  = x[a − bx − ],  dt 1 + x2 (3)   dy = y[ eα mx − β − δ y], dt 1 + x2 with x(0), y(0) > 0, and a, b, e, α , β , m are all positive constants. In the model, a is the intrinsic growth rate of the prey population; β is the intrinsic death rate of the predator population; b is strength of intra-specific competition among prey species; δ is strength of intra-specific competition among


339

predator species; m is direct measure of predator immunity from the prey; α is maximum attack rate of prey by predator and finally e represents the conversion rate. This system first was studied in [16], in which stability of its equilibria were studied analytically. We further study this system to reveal more dynamical behaviors and local bifurcations by employing numerical continuation method. We specially compute a curve of limit cycles, and several codim-1 bifurcation curves. The equilibrium of (3) are solutions of the following system:  α my  ] = 0,  x[a − bx − 1 + x2 (4) eα mx   y[ − β − δ y] = 0, 1 + x2 which are 1 ) The origin E0 = (0, 0). 2 ) E1 = ( ab , 0) is the axial equilibrium which always exists, as the prey population grows to the carrying capacity in the absence of predation. 3 ) E2 = (x2 , y2 ) is the positive equilibrium point which exists in the interior of the first quadrant if and only if there exists a positive solution to the following algebraic nonlinear equations. x2 = B5 x5 + B4 x4 + B3 x3 + B2 x2 + B1 x + B0 y2 = A3 x3 + A2 x2 + A1 x + A0 in which −b −b a −2b −2a d , B1 = 2 2 , , B4 = 2 2 , B3 = 2 2 , B2 = 2 2 + eα 2 m2 eα m eα m eα m eα m eα m d a , B0 = 2 2 + eα m eα m −b a −b a A3 = , A2 = , A1 = , A0 = . αm αm αm αm B5 =

2.1

Stability of E1

In this section, we analyze the local stability and bifurcation of the positive equilibrium E1 . The Jacobian matrix J of the model (3) evaluated at E1 is given by:   mα ab −a −  a2 + b2  J|E1 =  . emα ab − 0 − 2 β a + b2 The characteristic equation is given by

λ 2 − tr(J|E1 )λ + det(J|E1 ) = 0, Then the solutions of the characteristic gives the dispersion relation as q 1 λ1,2 = tr(J|E1 ) ± (tr(J|E1 ))2 − det(J|E1 ). 2 It is easy to see that the determinant of J|E1 is det(J|E1 ) = −

emα a2 b + aβ . a2 + b2

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The trace of J|E1 is tr(J|E1 ) = −a +

emα ab −β. a2 + b2

If condition tr(J|E1 ) < 0, det(J|E1 ) > 0 holds, then the positive equilibrium E1 of model (3) is a locally asymptotically stable node; if condition tr(J|E1 ) > 0, det(J|E1 ) < 0 holds, then E1 becomes a saddle. Hence, we have the following result. α ab , then the positive equilibrium E1 is locally asymptotical stable. Lemma 1. (i) If β + a > em a2 +b2 emα ab (ii) If β < a2 +b2 , then the positive equilibrium E1 is a saddle.

2.2

Stability and bifurcation of E2

We linearise the system (3) about the equilibrium point E2 . The Jacobian matrix J evaluated is   α my(1 − x2 ) α mx −  a − 2bx − (1 + x2 )2  1 + x2 . J|E2 =  2   emα x eα my(1 − x ) − − β − 2 δ y (1 + x2 )2 1 + x2 Theorem 2. Define b∗ =

2δ y2 x2

mx2 + 2xβ2 − 2xeα(1+x 2) + 2

2

α my2 (1+x22 ) 2x2 (1+x22 )2

then the following statements hold:

( i ) If b < b∗ , the positive equilibrium E2 is locally asymptotically stable . ( i i ) If b > b∗ , the positive equilibrium E2 is unstable. ( i i i ) If b = b∗ , then a supercritical Hopf bifurcation occurs around the positive equilibrium E2 . Proof. The characteristic equation is given by

λ 2 − tr(J|E2 )λ + det(J|E2 ) = 0, with det(J|E2 ) = (a − 2bx2 )(−β − 2δ y2 ), and tr(J|E2 ) = a − 2bx2 −

α my2 (1 − x22 ) eα mx2 + − β − 2δ y2 , 1 + x2 (1 + x22 )2

Then the solutions of the characteristic gives the dispersion relation as 1 λ1,2 = tr(J|E2 ) ± 2

q (tr(J|E2 ))2 − det(J|E2 ).

If det(J|E2 ) > 0 and tr(J|E2 ) < 0 then E2 is locally asymptotically stable. This implies b>

eα mx2 β α my2 (1 + x22 ) 2δ y2 + − + . x2 2x2 2x2 (1 + x22 ) 2x2 (1 + x22 )2

b
0,

Z. Lajmiri et al. / Journal of Applied Nonlinear Dynamics 7(4) (2018) 337–348 mx2 Suppose b∗ = 2δx2y2 + 2xβ2 − 2xeα(1+x 2) + 2

2

α my2 (1+x22 ) 2x2 (1+x22 )2

341

when b = b∗ ,tr(J|E2 ) = 0 and the characteristic equation has

p a pair of imaginary eigenvalues λ1,2 = ±i det(J|E2 ). Let λ (b) = v(b) ± iw(b) be the roots characteristic equation when b is near b∗ , then v(b) = 12 tr(J|E2 ) and d dv = Re(λ (b))|b=b∗ = −x2 < 0. db db when x∗ > 0. From the Poincare-Andronov-Hopf bifurcation theorem (See [2, 17, 18]), the model (3) undergoes a supercritical Hopf bifurcation at (b∗ , E2 ) when b = b∗ . Theorem 3. Define a∗ = 2bx2 +

α my2 (1−x22 ) (1+x22 )2

α mx2 − e1+x 2 + β + 2δ y2 then the following statements hold: 2

( i ) If a < a∗ , the positive equilibrium E2 is locally asymptotically stable . ( i i ) If a > a∗ , the positive equilibrium E2 is unstable. ( i i i ) If a = a∗ , then a subcritical Hopf bifurcation occurs around the positive equilibrium E2 . Proof. The proof is similar to that of the Theorem 2. Let λ (a) = p(a)± iq(a) be the roots characteristic equation when a is near a∗ , then p(a) = 12 tr(J|E2 ) and dp d = Re(λ (a))|a=a∗ = 1 > 0. da da 2.3

The cusp bifurcation analysis of E2

In this section, we discuss the cusp bifurcation of the system (3) (see, [2, 19] for more details). We transform the interior equilibrium point (x∗2 , y∗2 ) to the origin by setting x1 = x − x∗2 , x2 = y − y∗2 and then expand the system (3) in a power series around the origin, to obtain  dx   = a1 x1 + b1 x2 + p11 x21 + p12 x1 x2 + p22 x22 + O(kxk3 ), dt (5)   dy = c1 x1 + d1 x2 + q11 x2 + q12 x1 x2 + q22 x2 + O(kxk3 ), 1 2 dt ∂f ∂f ∂g ∂g ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∂ x |(x2 ,y2 ) , b1 = ∂ y |(x2 ,y2 ) , c1 = ∂ x |(x2 ,y2 ) , d1 = ∂ y |(x2 ,y2 ) , p11 2g 2g 2g ∂ ∂ ∂ 1 ∂2 f 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2 ∂ y2 |(x2 ,y2 ) , q11 = 2 ∂ x2 |(x2 ,y2 ) , q12 = ∂ x∂ y |(x2 ,y2 ) , q22 = 2 ∂ y2 |(x2 ,y2 ) , and

where a1 =

=

1 ∂2 f ∗ ∗ 2 ∂ x2 |(x2 ,y2 ) ,

p12 =

∂2 f ∗ ∗ ∂ x∂ y |(x2 ,y2 ) ,

α my2 α mx2 2x22 y2 α m 2x2 eα my2 + , b1 = − , c1 = − , 2 2 2 2 (1 + x2 ) (1 + x2 ) (1 + x2 ) (1 + x22 )2 eα m x2 y2 α m 2x2 y2 α m 2x22 y2 α m d1 = β − 2 δ y , p = −b + − + − , 2 11 (1 + x22 ) (1 + x22 )2 (1 + x22 )2 (1 + x22 )3 αm 2x22 α m eα my2 4x22 y2 eα m + , p = 0, q = − + , p12 = − 22 11 (1 + x22 ) (1 + x22 )2 (1 + x22 )2 (1 + x22 )3 2x2 eα m , q22 = −δ . q12 = − (1 + x22 )2

a1 = a − 2bx2 −

Then we must have a1 + d1 = 0, a1 d1 + b1 c1 = 0.

p22 =

342


We now use the following transformation y1 = x1 , y2 = a1 x1 + b1 x2 . Then model (5) reduces to

where

 dy 1  = y2 + α11 y21 + α12 y1 y2 + α22 y22 + O(kyk3 ),  dt   dy2 = β y2 + β y y + β y2 + O(kyk3 ), 11 1 12 1 2 22 2 dt

α11 =

(6)

p22 a21 p12 2p22 a1 p12 p22 − + + p11 , α12 = − , α22 = 2 , 2 2 b1 b1 b1 b1 b1

β11 = b1 q11 + a1 (p11 − q12 ) − β12 = −[2

a21 (p12 − q22 ) p22 a31 + 2 , b1 b1

p22 a21 a21 (p12 − 2q22 ) p22 a1 q22 − + − q12 ], β22 = . 2 b1 b1 b1 b21

There exists an C∞ invertible transformation given by 1 a1 p22 p12 + q22 2 p22 )y1 − 2 y1 y2 , z1 = y1 + ( 2 − 2 b1 b1 b1 z2 = y2 + (

a21 p22 a1 p12 a1 p22 q22 − + p11 )y21 − ( 2 + )y1 y2 . 2 b1 b1 b1 b1

Such that model (6) reduces to  dz1  = z2 + O(kzk3 )),  dt   dz2 = ρ z2 + ρ z z + O(kzk3 ), 1 1 2 1 2 dt

(7)

where ρ1 = β11 , ρ2 = − ba11 (p12 + 2q22 ) + 2p11 + q12 . If ρ1 ρ2 6= 0 (non-degeneracy condition), then the interior equilibrium point (x∗ , y∗ ) of the model (3) undergoes a cusp of codimension 2. Hence we derive the following theorem. Theorem 4. System (3) has a cusp point of codimension 2 at the equilibrium E2 .

3 Numerical simulation 3.1

Continuation curve of equilibrium points

The main aim of this section is to study the pattern of bifurcation that takes place as we vary the parameters a and b. This is actually done by studying the change in the eigenvalue of the Jacobian matrix and also following the continuation algorithm. To this end, we consider a set of fixed point initial solution, (x∗ , y∗ ) = (9, 3), corresponding to the parameter set of values, a = 4, b = 0.9, β = 0.01, δ = 0.01, e = 0.75, m = 0.75, α = 0.5. The characteristics of Hopf point, the limit cycle and the general bifurcation may be explored. To compute a curve of equilibria from (x∗ , y∗ ), we take a as the free parameter.


343

8.5

6.3

8 7.5

6.25 y

7 6.5 y

6.2

6 5.5

6.15

5 4.5

6.1

0.25

x

0.3

0

0.35

Fig. 1 Trajectories of system (3), when a = 2.42. E2 is locally asymptotically stable.

0.5

1

1.5

x

2

Fig. 2 Trajectories of system (3), when a = 2.41. E2 is unstable.

6

0.5 5

0.45

4

0.4

3 x

0.35 x

H

1

0.25

H

LP

2 a

4

0

0.2

Ŧ1

0.15 0

LP

2

0.3

20

40 t

60

80

100

Fig. 3 Hopf bifurcation occurs at E2 and bifurcating periodic solution for system (3) with a = 2.41, a = 2.42.

Ŧ2 Ŧ2

0

6

Fig. 4 Bifurcation diagram for the prey system (3) with intrinsic growth rate a in the presence of predator when a = 2.429.

From Figs. (1), (2), (3), (4) it is evident that the system has a subcritical Hopf point (H) and two limit point (LP) as follows: label = LP, x = ( 1.972773 10.342219 2.568313 ) a = 5.561259e-002 label = LP, x = ( 0.664201 11.962210 3.710425 ) a = 4.805268e-002 label = H , x = ( 0.278121 6.260537 2.429451 ) First Lyapunov coefficient = 9.292487e-002 We now compute a curve of equilibria by taking b as the free parameter with the same fixed set of

344


9.95 11

9.9 10.5

9.85 9.8

y

y

10

9.75

9.5

9.7 9.65

9 0.2

0.4

0.6

x

0.8

1

9.6

1.2

Fig. 5 Trajectories of system (3), when b = 2.055. E2 is locally asymptotically stable.

0.4

0.45

0.5

x

0.55

Fig. 6 Trajectories of system (3), when a = 2.155. E2 is unstable.

1

4

0.8 3

0.6 x

x

2

0.4

LP

1

H

LP

H

0.2 0

0 0

20

40 t

60

80

100

Fig. 7 Hopf bifurcation occurs at E2 and bifurcating periodic solution for system (3) with b = 2.155, b = 2.055.

1

1.5

b

2

2.5

3

Fig. 8 Equilibrium curve bifurcation diagram for system (3) with b = 2.105.

parameters as before. From Figs. (5), (6), (7), (8), it is evident that the system has a supercritical Hopf point (H) and two limit point (LP) at: label = LP, x = ( 1.457420 12.120672 1.746297 ) a = 7.743716e-002 label = H, x = ( 1.251242 12.716516 1.711339 ) Neutral saddle label = LP, x = ( 0.706672 12.255534 1.322906 ) a = -6.221778e-002 label = H, x = ( 0.468145 9.799718 2.105578 ) First Lyapunov coefficient = -1.235971e-003


345

9 12

8 7

11

H LPC

6

LPC H H

y

y

10

5

9

4 8

3 2.4

2

2.2 a

2

1

x

2

1.8

1 x

b

1.6 0

0

Fig. 9 The family of limit cycles bifurcating from the subcritical Hopf point H: LPC is a fold bifurcation of the cycle.

3.2

2

Fig. 10 Limit cycles and bifurcations of limit cycles starting from the supercritical Hopf point .

Cycle continuation starting from the Hopf-point

By starting from the subcritical Hopf point in the one-parameter bifurcation diagram of the equilibrium, we compute a curveof limit cycles, as depicted in Fig. (9) Limit point cycle (period = 2.681880e+001, parameter = 2.057916e+000) Normal form coefficient = -3.102923e-001 We now selecting supercritical Hopf point in the one-parameter bifurcation diagram of the equilibrium as initial point, and compute a curve of limt cycles, which is depicted in in Fig. (10) Limit point cycle (period = 1.408868e+001, parameter = 2.105578e+000) Normal form coefficient = -4.134387e-003 3.3

Continuation of the Hopf bifurcation

By taking subcritical Hopf point in the one-parameter bifurcation diagram of the equilibrium as initial point, and a, b as the free parameters with fixed β = 0.01, δ = 0.01, e = 0.75, m = 0.75, we can compute a curve of limit cycles. The Bogdanov-Takens are common points for the limit pont curves and corresponding to equilibria with eigen values λ1 + λ2 = 0. Actually, at each BT point, the Hopf bifurcation curve (with λ1,2 = ±iω0 , ω0 ) turns into the neutral saddle curve (with real λ1 = −λ2 ). Thus, we can start a Hopf curve from a Bogdanov-Takens point Fig. (11). label = BT, x = (0.953766 13.046759 4.872306 2.422320 0.000001) (a, b) = (5.135920e-002, -1.687727e+000) label = GH, x = (0.466980 9.782477 3.992203 2.099728 0.198890) l2 = -1.271506e-001 label = BT, x = (0.035601 -0.000003 -0.000001 0.000001 -0.000000) (a, b) = (-5.704058e-009, -2.801813e-001)

346


3 BT

2

GH

b

1 BT

0

Ŧ1

0

2

4

a

6

Fig. 11 Hopf curve in model (3), two-parameter bifurcation diagram.

13.06

13.06

13.05

13.055 13.05

y

y

13.04

13.045

13.03 13.04

13.02 13.035

13.01

0.92

0.94 x 0.96

0.98

1

Fig. 12 The equilibrium E1 is asymptotically stable for system (3).

3.4

13.03 0.93

0.94

0.95 x 0.96

0.97

Fig. 13 The equilibrium E1 is unstable for system (3).

Continuation of the fold bifurcation

By selecting fold point in the one-parameter bifurcation diagram of the equilibrium as initial point, and taking a, b as the free parameters with fixed β = 0.01, δ = 0.01, e = 0.75, m = 0.75, we can plot the one-parameter bifurcation curve of fold indicated in Figs. (12), (13), (14), (15) label = CP, x = ( 1.012217 13.061463 4.899576 2.450352 ) c = -1.965585e+000 label = BT, x = ( 0.953768 13.046761 4.872301 2.422315 ) (a,b) = (5.136031e-002, -1.687734e+000) label = CP, x = ( 0.035585 -0.000431 -0.373579 -10.493616 ) c = -6.335683e-005 label = CP, x = ( 0.005924 -0.833395 -0.375004 -10.549097 ) c = 1.857337e-005


347

5

1.5

CP BT

0

Ŧ5

x

b

1

BT

Ŧ10

CP CP

0.5

0

200

400 t

600

800

Fig. 14 The equilibrium E1 is asymptotically stable for system (3) and The equilibrium E1 is unstable for system (3).

Ŧ15

0

2 a

4

6

Fig. 15 fold curve in model (3) with E1 : BT Bogdanov-Takens points; CP - cusp.

4 Conclusion In this paper, we have investigated the complex dynamics of a predator-prey system with Holling type IV functional response. We have also investigated the distribution of the roots of the characteristic equation of the linearized system of the spatial model at the steady-state solution and discussed its stability. It has been shown that the system under consideration can undergo Hopf, cusp and BogdanovTakens bifurcations under certain conditions. We further compute a curve of limit cycles emanating from a Hopf bifurcation. For all detected bifurcations we numerically commuted the corresponding normal form coefficients. The numerical solutions and simulations are also given to verify the feasibility of the results. References [1] Gao, Y. (2013), Dynamics of a ratio-dependent predator-prey system with a strong Allee effect, Discrete and continuous dynamical systems series B, 18(9), 2283-2313. [2] Kuznetsov, Y. (1998), Elements of applied bifurcation theory, 112, Springer Verlag. [3] Agarwal, M. and Pathak, R. (2012), Harvesting and Hopf bifurcation in a prey-predator model with Holling type IV functional response, International Journal of Mathematics and Soft Computing, 2(1), 83-92. [4] Aziz-Alaoui, M.A. and Daher Okiye, M. (2003), Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16, 1069-1075. [5] Camara, B.I. and Aziz-Alaoui, M.A. (2008), Complexity in a prey predator model, International conference in honor of Claude Lobry, 9, 109-122. [6] Chen, S., shi, J., and wei, J. (2013), The effect of delay on a diffusive predator-prey system with Holling type-II predator functional response, Communications on pure and applied analysis, 12(1), 481-501. [7] Lu, H. and Wang, W. (2000), Application of fractional calculus in physics, world scientific. [8] Mukherjee, D., das, P., and kesh, D., Dynamics of a plant-herbivore model with Holling type II functional response, J. Math. [9] Zhang, L., Wang, W., Xue, Y., and Jin, Z. (2008), Complex dynamics of a Holling-type IV predator-prey model, 1-23. [10] Zhang, Z.Z and Yang H.Z. (2013), Hopf bifurcation in a delayed predator-prey system with modified Leslie-

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Gower and Holling type III schemes, Acta Automatica Sinica, 39(5), 610-616. [11] Banerjee, M. and Banerjee, S. (2012), Turing instabilities and spatiotemporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci, 236, 64-76. [12] Holling, C.S. (1965), The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Entomol. Sec. Can., 45, 1-60. [13] Murray, J.D. (1989), Mathematical Biology, Springer, Berlin. [14] Skalski, G. and Gilliam, J.F. (2001), Functional responses with predator interference: viable alternatives to the Holling type II model, Ecology, 82, 3083-3092. [15] Ruan, S. and Xiao, D. (2001), Global analysis in a predator-prey system with nonmonotonic functional response, Society for industrial and applied mathematics, 61(4), 1445-1472. [16] Vijaya Lakshmi, G.M., Vijaya, S., and Gunasekaran, M. (2014), Bifurcation and stability analysis in dynamics of prey-predator model with holling type IV functional response and intra-specific competition, International Journal Of Engineering And Science, 4, 52-61. [17] Perko, L. (2000), Differential Equations and and Dynamical Systems, New York, Springer-Verlag. [18] Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, Verlag New York Berlin Heidelberg. [19] Chow, S., Li, C., andWang, D. (2011), Dynamics of a delayed discrete semi-ratio dependent predator-prey system with Holling type IV functional response, Advances in Difference Equations, 3-19. [20] Agiza, H.N., Elabbasy, E.M., Metwally, E.K., and Elsadany, A.A. (2003), Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16, 1069-1075. [21] Allgower, E.L. and Georg, K. (1990), Numerical Continuation Methods: An Introduction, Springer- Verlag, Berlin. [22] Wang, Y.H. (2009), Numerical algorithm based on Adomian decomposition for fractional differential equations, Computer and Mathematics with Applications, 57, 1627-1681.



Delay-Coupled Mathieu Equations in Synchrotron Dynamics Revisited: Delay Terms in the Slow Flow Alexander Bernstein1 , Richard Rand2† 1 2

Center for Applied Mathematics, Cornell University, Ithaca NY 14853, USA Dept. of Mathematics and Dept. of Mechanical and Aerospace Engineering, Cornell University Submission Info Communicated by A.C.J. Luo Received 24 May 2017 Accepted 23 July 2017 Available online 1 January 2019 Keywords Delay-differential equation Mathieu equation Coupled oscillators Synchrotron

Abstract In a previous work, we applied perturbation methods to a system of two delay-coupled Mathieu equations, resulting in a slow flow which contains delayed variables. This previous treatment involved a convenient approximation which involved replacing delay terms in the slow flow by non-delay terms. The current paper explores the effect of keeping delay terms in the slow flow with the hope of illustrating what is lost in making such an approximation. Analytic results are shown to compare favorably with numerical integration of the slow flow itself. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In this paper we investigate the dynamics of the following system of two coupled Mathieu equations: x¨ + (δ + ε cost)x + εγ x3 + ε μ x˙ = εβ (xd + yd ),

(1)

y¨ + (δ + ε cost)y + εγ y + ε μ y˙ = εβ (xd + yd ) + εα x.

(2)

3

Here, xd ≡ x(t−T ) and yd ≡ y(t−T ). T is a constant which we will refer to as the delay of the system, and any term containing T in its argument, e.g. xd , we will refer to as a delay term. Coupled Mathieu equations without delay have been studied previously [1, 2]. In our previous paper [3], we explored the effect of including delay in eqs. (1), (2), but used a convenient approximation to replace the delay terms in the resulting slow flow with non-delay terms. Recent research [4,5] suggests that keeping the delay terms in the slow flow can result in more accurate analytic solutions. In this paper, we will use the method of two-variable expansion to derive the slow flow of the system. We will investigate both the case when delay terms are replaced with non-delay terms and the case when the delay terms are kept in the slow flow, and compare these results with each other as well as numerical integration of the slow flow. Percent error calculations will then be used to measure the error associated with approximating the delay terms with non-delay terms. † Corresponding

author. Email address: [email protected] ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2018.12.003

350

1.1

Alexander Bernstein, Richard Rand / Journal of Applied Nonlinear Dynamics 7(4) (2018) 349–360

Application

Motivation for studying this system comes from the field of accelerator physics [6]. A synchrotron is a type of particle accelerator that uses magnets arranged periodically in a circle to maintain the velocity of the charged particles moving along the circular trajectory. The charged particles are in collections, called bunches, and they leave wake fields behind them that can persist through entire orbits. The system (1), (2) models the dynamics of two bunches, where the parametric excitation represents the forcing of the magnets, the α terms represent the coupling between one bunch and the next, and the β terms represent the delayed self-feedback of the bunches as they pass through their wake fields from one orbit ago.

2 Two-variable expansion We use the two-variable expansion method [7, 8] to study the dynamics of eqs. (1), (2). We set

ξ (t) = t,

η (t) = ε t,

where ξ is the time t and η is the slow time. Since x and y are functions of ξ and η , the derivative with respect to time t is expressed through the chain rule: x˙ = xξ + ε xη , y˙ = yξ + ε yη . Similarly, for the second derivative we obtain: x¨ = xξ ξ + 2ε xξ η + ε 2 xηη ,

y¨ = yξ ξ + 2ε yξ η + ε 2 yηη .

In this paper we only perturb up to O(ε ), and so we will ignore the ε 2 terms. We then expand x and y in a power series in ε : x(ξ , η ) = x0 (ξ , η ) + ε x1 (ξ , η ) + O(ε 2),

y(ξ , η ) = y0 (ξ , η ) + ε y1 (ξ , η ) + O(ε 2).

(3)

In addition, we detune off of the 2:1 subharmonic resonance by setting:

δ=

1 + εδ1 + O(ε 2 ). 4

(4)

Substituting (3), (4) into (1), (2) and collecting terms in ε , we arrive at the following equations: 1 x0,ξ ξ + x0 4 1 y0,ξ ξ + y0 4 1 x1,ξ ξ + x1 4 1 y1,ξ ξ + y1 4

= 0,

(5)

= 0,

(6)

= −2x0,ξ η − μ x0,ξ − δ1 x0 − x0 cos ξ − γ x30 + β (x0d + y0d ),

(7)

= −2y0,ξ η − μ y0,ξ − δ1 y0 − y0 cos ξ − γ y30 + β (x0d + y0d ) + α x0 .

(8)

The solutions to (5) and (6) are simply:

ξ ξ x0 = A(η ) cos( ) + B(η ) sin( ), 2 2 ξ ξ y0 = C(η ) cos( ) + D(η ) sin( ). 2 2

(9) (10)


351

We then substitute (9), (10) into (7), (8). Note that:

ξ T ξ T x0d = A(η − ε T ) cos( − ) + B(η − ε T) sin( − ), 2 2 2 2 ξ T ξ T y0d = C(η − ε T ) cos( − ) + D(η − ε T) sin( − ). 2 2 2 2

(11) (12)

Using trigonometric identities, these equations can be written in terms of cos ξ2 and sin ξ2 . We set the coefficients of such terms equal to zero in order to remove the secular terms and avoid resonance. This results in four delay-differential equations in four unknowns: 3γ B 2 T T 1 μ A = −β sin( )(Ad +Cd ) − β cos( )(Bd + Dd ) − A + (δ1 − )B + (A + B2 ), 2 2 2 2 4 μ 3γ A 2 T T 1 (A + B2 ), B = −β sin( )(Bd + Dd ) + β cos( )(Ad +Cd ) − B − (δ1 + )A − 2 2 2 2 4 3γ D 2 T T 1 μ (C + D2 ) − α B, C = −β sin( )(Ad +Cd ) − β cos( )(Bd + Dd ) − C + (δ1 − )D + 2 2 2 2 4 3γ C 2 T T 1 μ (C + D2 ) + α A. D = −β sin( )(Bd + Dd ) + β cos( )(Ad +Cd ) − D − (δ1 + )C − 2 2 2 2 4

(13) (14) (15) (16)

It is common in the literature [9] to employ the following approximation here: Ad = A(η − ε T ) = A(η ) − ε TA + O(ε 2 ) = A(η ) + O(ε ). We will briefly go over this case in the next section to demonstrate its results. However, the point of this paper is to keep the delay terms Ad , Bd , Cd , Dd in the slow flow and observe the difference in slow flow dynamics.

3 Analytic results The present authors are interested in analyzing the stability of the origin of the original system (1), (2), which also corresponds to the origin of the slow flow (13), (14), (15), (16). To help accomplish this goal, we will linearize the slow flow around the origin and analyze the stability of that system, since the stability of the linearized system will be the same as the stability of the nonlinear system. Thus, the slow flow becomes: T T 1 μ A = −β sin( )(Ad +Cd ) − β cos( )(Bd + Dd ) − A + (δ1 − )B, 2 2 2 2 T T 1 μ B = −β sin( )(Bd + Dd ) + β cos( )(Ad +Cd ) − B − (δ1 + )A, 2 2 2 2 T T 1 μ C = −β sin( )(Ad +Cd ) − β cos( )(Bd + Dd ) − C + (δ1 − )D − α B, 2 2 2 2 T T 1 μ D = −β sin( )(Bd + Dd ) + β cos( )(Ad +Cd ) − D − (δ1 + )C + α A. 2 2 2 2

(17) (18) (19) (20)

As eqs. (17), (18), (19), (20) are linear in A, B, C, and D, we know that the general solution will be a linear combination of exponential functions. Thus, for instance, A = C1 eλ η and its derivative becomes: A =

d (C1 eλ η ) = C1 λ eλ η . dη

In addition, the delay term becomes: Ad = A(η − ε T) = C1 eλ η −ελ T = C1 eλ η e−ελ T .

352


Substituting these expressions into eqs. (17), (18), (19), (20) and expressing the system in matrix form, we obtain: ⎤⎡ ⎤ ⎡ ⎤ −β ν S −β ν C −β ν S − μ2 − λ −β ν C + δ1 − 12 C1 0 ⎥ ⎢ C2 ⎥ ⎢ 0 ⎥ ⎢ β ν C − δ1 − 1 − β ν S − μ − λ β ν C − β ν S 2 2 ⎥⎢ ⎥ = ⎢ ⎥. ⎢ ⎣ −β ν S −β ν C − α −β ν S − μ2 − λ −β ν C + δ1 − 12 ⎦ ⎣ C3 ⎦ ⎣ 0 ⎦ C4 0 β νC + α −β ν S β ν C − δ1 − 12 −β ν S − μ2 − λ ⎡

(21)

Where S = sin( T2 ), C = cos T2 , ν = e−ελ T . Our goal is to compare two approaches: 1) Replacing delay terms by non-delay terms, e.g. Ad by A, versus 2) Analyzing the system with the delay terms. Replacing delayed terms with non-delayed terms is equivalent to setting ν = 1, and this can be achieved by setting ε = 0 in the slow flow. To obtain nontrivial solutions we set the determinant of this matrix equal to zero, resulting in a characteristic equation of the form:

λ 4 + pλ 3 + qλ 2 + rλ + s = 0,

(22)

p = 4β ν S + 2 μ ,

(23)

where

q = 6β μν S − 4β δ1 ν C + 2αβ ν C + 4β 2 ν 2 +

3μ 2 1 + 2δ12 − , 2 2

r = 4β δ12 ν S + 4αβ δ1 ν S + 3β μ 2 ν S − β ν S − 4β δ1 μν C + 2αβ μν C + 4β 2 μν 2 +

s = 2β δ12 μν S + 2αβ δ1 μν S +

(24)

μ3 μ + 2δ12 μ − , (25) 2 2

β μ 3 ν S β μν S − 2 2

αβ μ 2 ν C αβ ν C − 2 2 4 2 1 μ δ μ2 μ2 δ2 + 1 − + δ14 − 1 + . +4β 2 δ12 ν 2 + 4αβ 2 δ1 ν 2 + β 2 μ 2 ν 2 + α 2 β 2 ν 2 − β 2 ν 2 + 16 2 8 2 16 −4β δ13 ν C − 2αβ δ12 ν C − β δ1 μ 2 ν C + β δ1 ν C +

(26)

Note that p, q, r, and s all depend on λ appearing in exponential form, so eq. (22) is not a polynomial equation. In the previous paper the Routh-Hurwitz criterion [10] was used to determine the stable regions of the characteristic polynomial, but since eq. (22) is a transcendental equation a different technique will have to be used here. A necessary condition for stability is for the real part all of the eigenvalues λi to be nonpositive, and the transition curves between stable and unstable regions in parameter space occur when the real part of λi is exactly zero. The approach we’ll take to find the transition curves is to set λ = iω ; the case when ω = 0 corresponds to a saddle node bifurcation, and all nonzero values of ω correspond to a possible Hopf bifurcation in the nonlinear system. Substituting λ = iω in (22) turns it into a complex equation, and to solve it we set the real and imaginary parts equal to zero: 0 = 4β 2 μω sin (2εω T ) − 4β 2 ω 2 cos (2εω T ) + 4β 2 δ12 cos (2εω T ) + 4αβ 2 δ1 cos (2εω T ) + β 2 μ 2 cos (2εω T ) +α 2 β 2 cos (2εω T ) − β 2 cos (2εω T ) − 4β ω 3 S sin (εω T ) + 4β δ12 ω S sin (εω T ) + 4αβ δ1 ω S sin (εω T ) +3β μ 2 ω S sin (εω T ) − β ω S sin (εω T ) − 4β μδ1 ω C sin (εω T ) + 2αβ μω C sin (εω T ) β μ 3 S cos (εω T ) −6β μω 2 S cos (εω T ) + 2β μδ12 S cos (εω T ) + 2αβ μδ1 S cos (εω T ) + 2


353

β μ S cos (εω T ) + 4β δ1 ω 2 C cos (εω T ) − 2αβ ω 2 C cos (εω T ) − 4β δ13 C cos (εω T ) 2 αβ μ 2 C cos (εω T ) −2αβ δ12 C cos (εω T ) − β μ 2 δ1 C cos (εω T ) + β δ1 C cos (εω T ) + 2 2ω 2 2 2δ 2 2 4 3 1 αβ C cos (εω T ) μ ω μ δ μ μ2 1 − + ω 4 − 2δ12 ω 2 − + + δ14 + − 1 + − + (27) 2 2 2 2 2 16 8 16 0 = 4β 2 ω 2 sin (2εω T ) − 4β 2 δ12 sin (2εω T ) − 4αβ 2 δ1 sin (2εω T ) − β 2 μ 2 sin (2εω T ) − α 2 β 2 sin (2εω T ) +β 2 sin (2εω T ) + 4β 2 μω cos (2εω T ) + 6β μω 2 S sin (εω T ) − 2β μδ12 S sin (εω T ) − 2αβ μδ1 S sin (εω T ) β μ 3 S sin (εω T ) β μ S sin (εω T ) + − 4β δ1 ω 2 C sin (εω T ) + 2αβ ω 2 C sin (εω T ) + 4β δ13 C sin (εω T ) − 2 2 αβ μ 2 C sin (εω T ) αβ C sin (εω T ) + +2αβ δ12 C sin (εω T ) + β μ 2 δ1 C sin (εω T ) − β δ1 C sin (εω T ) − 2 2 −4β ω 3 S cos (εω T ) + 4β δ12 ω S cos (εω T ) + 4αβ δ1 ω S cos (εω T ) + 3β μ 2 ω S cos (εω T ) μ 3 ω μω −β ω S cos (εω T ) − 4β μδ1 ω C cos (εω T ) + 2αβ μω C cos (εω T ) − 2μω 3 + 2μδ12 ω + − . (28) 2 2 −

These equations are very messy, and it is not clear if a closed form solution T (δ1 , α , β , μ , ε ) can be found by eliminating ω . To simplify matters, we will start by setting ε = 0 in eqs. (27), (28), which is equivalent to approximating the delayed terms in the slow flow as non-delayed terms. 3.1

The non-delayed case

Setting ε = 0 in eqs. (27), (28) yields: 0 = −4β 2 ω 2 + 4β 2 δ12 + 4αβ 2 δ1 + β 2 μ 2 + α 2 β 2 − β 2 − 6β μω 2 S + 2β μδ12 S + 2αβ μδ1 S β μ 3S β μ S − + 4β δ1 ω 2 C − 2αβ ω 2 C − 4β δ13 C − 2αβ δ12 C − β μ 2 δ1 C + β δ1 C + 2 2 1 3μ 2 ω 2 ω 2 αβ μ 2 C αβ C μ 2 δ12 δ12 μ 4 μ 2 − + ω 4 − 2δ12 ω 2 − + + δ14 + − + − + , + 2 2 2 2 2 2 16 8 16 2 2 3 2 2 0 = 4β μω − 4β δ1 ω C − 4β ω S + 4β δ1 ω S + 4αβ δ1 ω S + 3β μ ω S − β ω S μ 3 ω μω − . −4β μδ1 ω C + 2αβ μω C − 2μω 3 + 2μδ12 ω + 2 2

(29)

(30)

This system is simple enough to have a closed form solution. We start by solving eq. (30) for ω , resulting in the trivial solution: ω = 0, and the nontrivial solution: 8β δ12 + 8αβ δ1 + 6β μ 2 − 2β S + (4αβ μ − 8β μδ1 ) C + 4μδ12 + μ 3 + 8β 2 − 1 μ 2 ω = . 8β S + 4μ

(31)

The saddle node transition curves are obtained by substituting ω = 0 in eq. (29): 0 = 32β μδ12 + 32αβ μδ1 + 8β μ 3 − 8β μ S + 16β δ1 − 64β δ13 − 32αβ δ12 − 16β μ 2 δ1 + 8αβ μ 2 − 8αβ C

+16δ14 + 8μ 2 δ12 + 64β 2 δ12 − 8δ12 + 64αβ 2 δ1 + μ 4 + 16β 2 μ 2 − 2μ 2 + 16α 2 β 2 − 16β 2 + 1.

The Hopf transition curves are obtained by substituting (31) in eq. (29):

(32)

354


0 = 128αβ 3 μδ1 S 3 + 64β 3 μ 3 S 3 − 16α 2 β 3 μ S 3 − 16β 3 μ S 3 − 128β 3 μ 2 δ1 C S 2 + 32α 2 β 3 δ1 C S 2 +64αβ 3 μ 2 C S 2 + 64β 2 μ 2 δ12 S 2 − 16α 2 β 2 δ12 S 2 + 32αβ 2 μ 2 δ1 S 2 + 48β 2 μ 4 S 2 + 128β 4 μ 2 S 2 −4α 2 β 2 μ 2 S 2 − 20β 2 μ 2 S 2 − 16α 2 β 4 S 2 + 64αβ 2 μδ12 C S − 64β 2 μ 3 δ1 C S − 128β 4 μδ1 C S +32αβ 2 μ 3 C S + 64αβ 4 μ C S + 32β μ 3 δ12 S + 64β 3 μδ12 S − 128αβ 3 μδ1 S + 12β μ 5 S +64β 3 μ 3 S − 8β μ 3 S + 64β 5 μ S + 16αβ μ 2 δ12 C − 8β μ 4 δ1 C − 32β 3 μ 2 δ1 C + 4αβ μ 4 C +16αβ 3 μ 2 C + 4μ 4 δ12 + 16β 2 μ 2 δ12 − 32αβ 2 μ 2 δ1 + μ 6 + 8β 2 μ 4 − μ 4 + 16β 4 μ 2 .

(33)

Figure 1 shows both the saddle node bifurcation curves and the Hopf bifurcation curves for the nondelayed system, as well as the stable regions. Stable regions are determined by selecting a representative point from each disjoint region and testing that point for stability. We now return to eqs. (27), (28) and employ a perturbation approach to calculate the Hopf bifurcation. 3.2

The delayed case

Since we now have a solution when ε = 0, the next reasonable course of action would be to look for a series solution in ε with our result being the zeroth order solution. Unfortunately, the result when ε = 0 is still too complicated to be written explicitly in a closed form solution. In order to determine the effect of ε on the system, we also need to perturb off of α and μ , resulting in a series expansion in three variables. Fortunately, the saddle node bifurcations are the same for both the delayed and non-delayed systems, so we do not need to investigate the case when ω = 0 as this solution is already known exactly. We begin by expanding T and ω in the following series: T = T000 + T100 α + T010ε + T001 μ + T200 α 2 + T110αε + T101 α μ + T020 ε 2 + T011 ε μ + T002 μ 2 + · · ·

(34)

ω = ω000 + ω100 α + ω010 ε + ω001 μ + ω200 α 2 + ω110 αε + ω101 α μ + ω020 ε 2 + ω011 ε μ + ω002 μ 2 + · · · (35) We first calculate the zeroth order terms T000 and ω000 and use those results to calculate higher order terms. Substituting (34), (35) into eqs. (27), (28) and setting α = 0, ε = 0, and μ = 0, eqs. (27), (28) become: 2 2 − 4δ 2 − 16β 2 + 1 + 4ω000 4ω000 − 4δ12 + 1 16β δ1 cos T000 1 2 , (36) 0 = 16 2 T000 ). (37) − 4δ12 + 1 sin( 0 = −β ω000 4ω000 2 2 − 4δ 2 + 1 = 0; Looking at eq. (37), we see there are three distinct cases to examine: ω000 = 0; 4ω000 1 T000 and sin( 2 ) = 0. As we are not interested in the case when ω = 0, we will only focus on the second and third cases. 2 − 4δ 2 + 1 = 0, eq. (36) is also identically zero. Thus, this single In the second case when 4ω000 1

condition satisfies both eq. (36) and eq. (37). In this case, we get ω000 = δ12 − 14 , which undergoes a Hopf bifurcation when |δ1 | > 1/2. In the third case when sin( T000 2 ) = 0, the condition is satisfied when T000 = 2nπ for integer values of n. In this paper we will examine the smallest nonzero delay at which a Hopf bifurcation occurs, which is when T000 = 2π .


355

Fig. 1 The top graph shows the transition curves for the saddle node bifurcations of the non-delayed system (ε = 0). The bottom graph shows the transition curves for the Hopf bifurcations of the non-delayed system (ε = 0). The middle graph shows both sets of transition curves and has the stable regions shaded in. Parameter values are: α = 0.1; β = 0.125; μ = 0.01.

356


Substituting T000 = 2π into (36) yields the expression: 2 − 4δ12 − 16β 2 + 1 = 0, −16β δ1 + 4ω000

Solving for ω000 gives us the solution:

ω000 =

4δ12 + 16β δ1 + 16β 2 − 1 2

.

Since ω > 0 for a Hopf bifurcation, the radicand must be positive. The radicand is positive for δ1 < −2β −1/2 and δ1 > −2β +1/2, which fully determines the Hopf bifurcation for the case of T000 = 2π . This means that the Hopf curve represents a necessary condition for a Hopf bifurcation, but not a sufficient one. The points where the Hopf curve intersect the saddle node curves divide the Hopf curve into disjoint regions; the additional conditions δ1 < −2β − 1/2 and δ1 > −2β + 1/2 are used to determine which of those disjoint regions represent actual Hopf bifurcations. The third case is the solution we will focus on in this paper, and so we will pick T000 = 2π and

ω000 = 4δ12 + 16β δ1 + 16β 2 − 1/2 as our zeroth order solutions in the perturbation method. By substituting the zeroth order solutions back into eqs. (27), (28), we are then able to solve for higher order terms. At each step in this process, ωi jk and Ti jk are solved simultaneously, just as ω000 and T000 were in the zeroth order case. The final result for (34), (35) is: T = 2π +

μ δ 1 + 2β ε μ + 8π (δ1 + 2β )2 ε 2 + HOT, − 4π (δ1 + 2β ) ε + 2παε − 2β β

3 1 1 (δ1 + 2β ) α − 4δ1 + 24β δ12 + 48β 2 δ1 − δ1 + 32β 3 − β α 2 3 2ω000 32β ω000 1 2 1 π − π β δ1 ε 2 + εμ − (δ1 + 2β ) μ 2 + HOT. ω000 4ω000 16β ω000

(38)

ω = ω000 −

(39)

Figure 3 shows eq. (38) and eq. (32). Since we are mostly concerned with the area around T = 2π , the graphs shown here will zoom in on that region. A blowup of Figure 1 is shown in Figure 2 for reference.

4 Numerical results The numerical computations use DDE23 in MATLAB [11] to numerically integrate the slow flow (13), (14), (15), (16). These numerical results will be compared to the analytical results presented earlier in the paper. We note that the analytical results are approximate due to the perturbation method, which truncates the solution, neglecting higher order terms; in this way both the numerical and the analytic approaches are approximate. Additionally, the slow flow only captures the behavior of the system on the slow time scale, and thus ignores some of the structure of the original system. We find much better agreement when comparing the analytical results to the numerical integration of the slow flow instead of the numerical integration of the original system, and this stems from the fact that the slow flow is itself only an approximation. We will only integrate the slow flow in this paper. The code we used determines stability of the origin by taking the initial condition as a point close to the origin and checking if the amplitude grows without bound. Since nonlinear terms will trap unstable trajectories in a limit cycle of finite amplitude, the techniques outlined here only work with the linearized system.


357

⇓

Fig. 2 The top graph is Figure 1. The bottom graph zooms in on the region for T between 5 and 8, while keeping the range of δ1 values the same.

We utilized a combination of two techniques to accomplish this goal: first, we select an upper bound on amplitude size and quit out of integration if that amplitude is reached; second, for all other cases we check if the maximum amplitude over the entire time interval is equivalent to the maximum amplitude over a subsection of the time interval. This latter method helps capture edge cases that the former doesn’t catch. In all computations we used α = 0.1, β = 0.125 and μ = 0.01. In Figure 4 we see the effect of including delay purely from the standpoint of numerical integration. The bottom graph in the figure also compares the analytic results to the results of numerical integration.

5 Conclusion In this paper, we investigated the dynamics of two coupled Mathieu equations with delay. In particular we analyzed the stability of the origin and the effect of delay and damping on stability. We used the method of two variable expansion to calculate a characteristic equation of the system’s slow flow, and used power series to analyze the Hopf bifurcations around T = 2π ; these results were then compared with numerical integration. The numerical results of the slow flow closely matched the analytical results for small values of ε , α and μ . Furthermore, within the range of parameter values for which agreement held, we found significant variation in the Hopf bifurcation transition curve. These results demonstrate that including delay terms in the slow flow can be very important when approximating numerical results.

358


Fig. 3 The top graph is the blowup of Figure 1. The bottom graph shows the series solutions for the Hopf curves for both the delayed and non-delayed systems, as well as the saddle node bifurcation curves.

To measure the significance of replacing delay terms by non-delay terms, we calculate the error between the value of T at the stability boundary obtained through numerical integration of the slow flow and the value of T obtained through the series solution, for both the entire series and just the zeroth order terms. The percent error is calculated as follows: error =

|Tnum − Tseries | × 100%. Tnum

For fixed values of δ1 between -1 and 1, different values of Tnum and Tseries are obtained; the Table below shows the average and maximum values of the error in this range. In the case of the maximum error, including delay terms results in a more accurate solution by a factor of 10. When looking at the average error, the accuracy is even better. This demonstrates what is lost by omitting the delay terms in the slow flow.

Acknowledgments The authors would like to thank their colleagues J. Sethna, D. Rubin, D. Sagan and R. Meller for introducing us to the dynamics of the Synchrotron and for their continued support in this research. This work was partially supported by NSF Grant PHY-1549132.

Alexander Bernstein, Richard Rand / Journal of Applied Nonlinear Dynamics 7(4) (2018) 349–360 Omitting delay terms (ε = 0)

Including delay terms (ε = 0.1)

Average

15.04%

0.60%

Maximum

19.34%

1.88%

359

Fig. 4 The top graph shows the stable points near the Hopf curve as computed through numerical integration of the slow flow, with the circles representing the non-delayed system and the asterisks representing the delayed system. The bottom graph shows the Hopf and saddle node transition curves on top of these stable regions.

References [1] Hsu, C.S. (1961), On a restricted class of coupled Hill’s equations and some applications, Journal of Applied Mechanics, 28(4), Series E, 551. [2] Bernstein, A. and Rand, R.H. (2016), Coupled Parametrically Driven Modes in Synchrotron Dynamics, Chapter 8, 107-112 in Nonlinear Dynamics, Volume 1: Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, G. Kerschen, editor, Springer. [3] Bernstein, A. and Rand, R. (2016), Delay-coupled mathieu equations in synchrotron dynamics, Journal of Applied Nonlinear Dynamics, 5(3), 337-348. [4] Sah, S.M. and Rand, R.H. (2016), Delay terms in the slow flow, Journal of Applied Nonlinear Dynamics, 5(4), 471-484. [5] Bernstein, A., Sah, S.M., Meller, R.E., and Rand, R.H. (2017), Hopf bifurcation in a delayed nonlinear Mathieu equation, Proceedings of 9th European Nonlinear Dynamics Conference (ENOC 2017), 25-30, Budapest, Hungary. [6] “Cornell Electron Storage Ring.” CLASSE: CESR. 2014 Cornell Laboratory for Accelerator-based Sciences

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[7] [8] [9] [10] [11]


and Education Kevorkian, J. and Cole, J.D. (1981), Perturbation Methods in Applied Mathematics, Applied Mathematical Sciences, 34, Springer. Rand, R.H. (2012), Lecture Notes in Nonlinear Vibrations, Published on-line by The Internet-First University Press, . Morrison, T.M. and Rand, R.H. (2007), 2:1 Resonance in the delayed nonlinear Mathieu equation, Nonlinear Dynamics, 341-352, DOI:10.1007/s11071-006-9162-5. Routh, E.J. (1877), A treatise on the stability of a given state of motion, particularly steady motion, London, U.K., Macmillan. MATLAB’s reference on dde23, http://www.mathworks.com/help/matlab/ref/dde23.html



A Computational and Theoretical Review on the Motion of a Spinning Spherical Particle in Media with Different Viscosities F. L. Braga†, I. G. Pauli, V. S. Amorim Coordenadoria de F´ısica, Instituto Federal de Educa¸cão, Ciências e Tecnologia do Esp´ırito Santo, Campus Cariacica, Av. José Sette s/n, Espirito Santo, 29150-410, Brasil Submission Info Communicated by A.C.J. Luo Received 9 January 2017 Accepted 9 August 2017 Available online 1 January 2019 Keywords Trajectory Magnus effect Computational simulation Drag force

Abstract Ballistic calculations were the first attempt where computers were used for, and the oblique launching of objects when studied at an ideal approach is quite simple with a parabolic trajectory of particles. The same motion could become intrinsically difficult to solve when dissipative forces are considered at the model. The present work shows a brief theoretical review on the motion of a spinning spherical particle under the influence of the gravitational field, the drag force and the Magnus effect. The drag forces can alter the translation speed and the angular velocity. We determined the profiles of the trajectories, velocity field and the modifications of frequency in two scenarios, first when the particle is moving across one medium and second when it pass through one medium to another. The results, trajectories obtained are in agreement the predictions for the cases where non continuous and non uniform forces are acting on a body. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The studies associated with the motion of objects under the influence of gravitational fields are one of the main tasks of kinematics. In regular physics courses for undergraduate students and at high school, the concepts of referential, position, trajectory, velocity, speed, acceleration and others are usually the primary physical ideas learned, as can be seen in the arrangement of a great amount of text books [1–5]. In general, from the perspective of complexity, these initial studies are simplified, in order to make the problem analytically treatable from the mathematical point of view, and the results obtained are ideal, but useful. The simplifications that were generally done is to avoid the influence of dissipative forces [1, 6, 7]. Approaches considering drag forces [7], turbulent flow [8] and the famous Magnus effect [8] makes the equations of motion [6, 7] that governs the particles trajectories, typical non-linear differential equations [9] that can conveniently be solved using numerical calculations techniques [10]. The importance associated with these studies is based on the applications of all results obtained. Big part of these results are intrinsic correlated to ballistic purposes [11, 12] and fluid flow research areas [13,14] commonly investigated in aeronautics, pharmacology, chemical and automobile industries. † Corresponding



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F. L. Braga, I. G. Pauli, V. S. Amorim / Journal of Applied Nonlinear Dynamics 7(4) (2018) 361–369

d

Fig. 1 Force diagram associated with the flight of the spherical object under the influence of the gravitational force, drag forces and Magnus effect.

It is possible to detect in literature works that analyzed Magnus effect and Drag forces influence on the motion of objects [15–19], but, there is no theoretical research associating this two dissipative effects and the modifications of the angular velocities of a particle under the influence of Magnus effect and fluid viscosity. Our present work numerically investigates the previous cited aspect and analyzed the trajectory of a spherical particle launched obliquely to the surface of Earth and that pass through one medium to another, simulating a discontinuity on the dissipative forces acting on the particle motion. The results obtained show a typical dependence on the particle original angular velocity and the trajectory generally shows a very distinct behaviour from the expected for high angular velocity (≈ 1200rad/s). 2 Model The problem investigated was the flight of a homogeneous solid spherical object of radius R similar to the one shown in Fig. 1, under the influence of the Earth’s gravitational field, drag forces due to the medium in which the object is immersed, and the Magnus Force. The drag forces must be responsible for the reduction of translational and rotational kinetic energy associated with the particle. A cartesian system of coordinates was considered. For simplicity the particle realizes a horizontal launch parallel to the x direction. It spins with an angular momentum ~L pointing on z direction and its center of mass present a velocity ~v initially pointed only on the x direction. The drag force modulus was proposed as 1 Fd = A pCd ρ v2 , 2

(1)

where A p is the transverse area of the projectile, ρ is the medium density and Cd is the drag coefficient, a dimensionless number that we consider constant Cd ≈ 0, 5 taking into account a perfect sphere as particle and v2 = (v2x + v2y + v2z ) the square modulus of velocity. The component i = {x, y, z} of the force


363

Fig. 2 Spherical project of radius spinning around the z axis.

was given by Fdi = −Fd q

vi

,

v2x + v2y + v2z

been vi the component of the velocity. Since the y coordinates is smaller when compared to the Earth’s radius, so the modulus of the gravitational force is given by |~Fg | = mg,

(2)

been g the gravity acceleration, approximately constant. Across the simulations, it was considered as g ≈ 9, 8m/s2 . The Magnus force can be cast as ~ω ×~v 1 ~ ), Fm = Cs A p ρ v2 ( ~ ×~v| 2 |ω

(3)

where Cs is the sustaining coefficient that depends on morphological characteristics of the analyzed object [15], ~ω the projectile angular velocity pointing parallel to the z axis, as depicted on Fig. 2, ρ is the medium density where the particle is emerging. For simplicity along the motion ~v = vx xˆ + vy y, ˆ and ~ = ω zˆ then, ~Fm was simplified as ω ~Fm = 1 Cs A p ρ v (vx yˆ − vy x) ˆ . (4) 2 The sustaining coefficient was related to the Magnus coefficient proposed as Cm = rotation parameter given by S =

(ω R) v .

Cs S−1 sen(φ ) ,

where S is the

Since ω = 2π f been f the rotation frequency, hence we have the

364


Table 1 Medium parameters Values used along the simulations (International System of units). Media

T (◦ C)

ρ ( mkg3 )

η (Pa · s)

Air

20

1.200

17, 200 × 10−6

Water

20

1.000

1, 0030 × 10−3

Magnus force given by ~ ˆ , Fm = π RA p f ρ (−vy xˆ + vx y)

(5)

for perfect smooth spheres we have that Cm ≈ 1. ~ in a According to the reference [20] a spinning spherical object of radius R with angular velocity ω medium with viscosity η is subjected to a torque ~τ = −8π R3 η ~ω .

(6)

Taking into account all these forces, we calculate the resultant force ~FR for rotational and translational coordinates. Adding equations (1), (2) and (5), we found m(ax xˆ + ay y) ˆ = π RA p f ρ (−vy xˆ + vx y) ˆ − mgyˆ − Fd q

vx v2x + v2y + v2z

xˆ − Fd q

τ zˆ = −8π R3 ηω zˆ.

vy

y, ˆ

(7)

v2x + v2y + v2z (8)

Here we neglected the buoyancy forces even when the particles are immersed in water, since, the approximate ratio between the modulus of buoyancy forces Fb and Magnus forces Fm , under the consideration here proposed (normalized sphere m = 1Kg, R = 1m, then the sphere density was given as ρsphere = 43π kg/m3 ) mg g Fb ≈ = 0, of h defined by ˆ t dn C α Dt h(t) = gn−α (t − s) n h(s)ds, ds 0 β −1

t where n is the smallest integer greater than or equal to α and gβ (t) = Γ( β ) ,t > 0, β ≥ 0. The history xt : (−∞, 0] → X given by xt (θ ) = x(t + θ ) belongs to some abstract phase space B defined axiomatically. Further, x : I × B × X → X, k : D × B → X : D = {(t, s) ∈ I × I : 0 ≤ s ≤ t ≤ a}, ρ : I × B → (−∞, T] are apposite functions. The paper is structured as follows: section 2 deals some basic definitions and preliminary facts. In section 3, existence of mild solution of the model (1)-(2) is discussed using Banach contraction technique. An example is offered in section 4.

2 Preliminaries Let L (X) symbolizes the Banach space of all bounded linear operators from X into X endowed with the uniform operator topology, having its norm recognized as · L (X) . Let C(I , X) symbolize the space of all continuous functions from I into X, having its norm recognized as · C(I ,X) . Moreover, Br (x, X) symbolizes the closed ball in X with the center at x and the distance r. Once the delay is infinite, it is necessary to consider the theoretical phase space Bh in a proper way. The phase space (Bh , · Bh ) is a semi - normed linear space of functions mapping (−∞, 0] into X. If x : (−∞, T] → X, T > 0, is continuous on I and x0 ∈ Bh , then for every t ∈ I , the accompanying conditions hold: P1 xt is in Bh ; P2 x(t)X ≤ Hxt Bh ; P3 xt Bh ≤ D1 (t)sup{x(s)X : 0 ≤ s ≤ t} + D2 (t)x0 Bh , where H > 0 is a constant and D1 (·) : [0, +∞) → [0, +∞) is continuous, D2 (·) : [0, +∞) → [0, +∞) is locally bounded, and D1 , D2 are independent of x(·). P4 The function t → ςt is well described and continuous from the set R(ρ − ) = {ρ (s, ς ) : (s, ς ) ∈ [0, T] × Bh}, into Bh and there is a continuous and bounded function J ς : R(ρ − ) → (0, ∞) to ensure that ςt Bh ≤ J ς (t)ς Bh , for every t ∈ R(ρ − ).

K. Jothimani et al. / Journal of Applied Nonlinear Dynamics 7(4) (2018) 371–381

373

Recognize the space BT = {x : (−∞, T] → X : x|I is continuous and x0 ∈ Bh }, where x|I is the constant of x to the real compact interval on I . The function · BT to be seminorm in BT , described by xBT = ς Bh + sup{x(s)X : s ∈ [0, T]},

x ∈ BT .

Lemma 1. (see [30]). Let x : (−∞, T] → X be a function in a way that x0 = ς , and if P4 hold, then xs Bh ≤ (D2∗ + J ς )ς Bh + D1∗ sup{x(θ )X : θ ∈ [0, max{0, s}]},

where J ς = sup J ς (t), t∈R(ρ − )

s ∈ R(ρ − ) ∪ I ,

D1∗ = sup D1 (s),

D2∗ = sup D2 (s).

s∈[0,T]

s∈[0,T]

For outcomes, we assume that abstract fractional integro-differential problem ˆ t α B(t − s)x(s)ds, Dt x(t) = Ax(t) +

(3)

0

x(0) = ϕ ∈ X,

x (0) = 0,

(4)

has associated α −resolvent operator of bounded linear operators (Tα (t))t≥0 on X. Definition 1. (see [28]). A one parameter family of bounded linear operators (Tα (t))t≥0 on X is called a α −resolvent operator of (3)-(4) if the following conditions are verified. (a) The function Tα (·) : [0, ∞) → L (X) is strongly continuous and Tα (0)x = x, for all x ∈ X and α ∈ (1, 2). (b) For x ∈ D(A), Tα (·)x ∈ C([0, ∞), [D(A)]) ∩C1 ([0, ∞), X), and ˆ t B(t − s)Tα (s)xds, (5) Dtα Tα (t)x = ATα (t)x + 0 ˆ t α Tα (t − s)B(s)xds, (6) Dt Tα (t)x = Tα (t)Ax + 0

for every t ≥ 0. The existence of a α −resolvent operator for problem (3)-(4) was studied in [30]. In this work we have considered the same conditions in P1-P3 which are same as stated in [29], consequently we preclude it. In view of P1-P3, in the sequel, for r > 0 and θ ∈ ( π2 , ϑ )

∑ = {λ ∈ C : λ = 0, |λ | > r, |arg(λ )| < θ }, r,θ

for Γr,θ , Γir,θ , i = 1, 2, 3 are the paths Γ1r,θ = {teiθ : t ≥ r}, Γ2r,θ = {reiε : −θ ≤ ε ≤ θ }, Γ3r,θ = {te−iθ : t ≥ r} and Γr,θ = ∪3i=1 Γir,θ oriented clockwise. Furthermore, ρα (Gα ) are the sets λ ))−1 ∈ L (X)}. ρα (Gα ) = {λ ∈ C : Gα (λ ) := λ α −1 (λ α I − A − B(

Presently, we determine the operator family (Tα (t))t≥0 by ˆ ⎧ ⎨ 1 eλ t Gα (λ )d λ , Tα (t) = 2π i Γr,θ ⎩ I,

t > 0, t = 0.

Now, we list out some conventional outcomes from current works.

(7)

374


Theorem 2. (see [31]). Assume conditions P1-P3 are fulfilled. Then there exists a unique α −resolvent operator for problem (3)-(4). Theorem 3. (see [31]). The function Tα : [0, ∞) → L (X) is strongly continuous and Tα : (0, ∞) → L (X) is uniformly continuous. We assume that the conditions P1-P3 are satisfied. Further, we need to talk about the mild solution of the model (1)-(2). For this intent, it is necessary to discuss the subsequent non-homogeneous model Dtα x(t)

ˆ

t

= Ax(t) +

B(t − s)x(s)ds + F (t),

t ∈I,

(8)

0

x(0) = ϕ ∈ X,

x (0) = 0,

(9)

where α ∈ (1, 2) and F ∈ L1 (I , X). In the follow up, Tα (·) is the operator function characterized by (7). Now, we start by presenting the subsequent concept of classical solution. Definition 2. (see [28]). A function x : I → X, 0 < T, is called a classical solution of (8)-(9) on I if x ∈ C(I , [D(A)]) ∩ C(I , X), μ n−α ∗ x ∈ C1 (I , X), n = 1, 2, the condition (9) holds and the equation (8) is verified on I . Definition 3. (see [28]). Let α ∈ (1, 2), we describe the family (Sα (t))t≥0 by ˆ Sα (t)x =

0

t

μ α −1 (t − s)Tα (s)ds,

for each t ≥ 0. Lemma 4. (see [31]). If the function Tα (·) is exponentially bounded in L (X), then Sα (·) is exponentially bounded in L (X). Lemma 5. (see [31]). If the function Tα (·) is exponentially bounded in L ([D(A)]), then Sα (·) is exponentially bounded in L ([D(A)]). Theorem 6. (see [31]) Let z ∈ D(A). Assume that F ∈ C(I , X) and x(·) is called a classical solution of (8) − (9) on I . Then ˆ x(t) = Tα (t)z +

t 0

Sα (t − s)F (s)ds,

t ∈I.

(10)

It is obvious from the earlier definition that Tα (·)z is a solution of problem (3)-(4) on (0, ∞) for z ∈ D(A). Definition 4. (see [28]). Let F ∈ L1 (I , X). A function x ∈ C(I , X) is called a mild solution of (8)-(9) if ˆ t Sα (t − s)F (s)ds, t ∈I. x(t) = Tα (t)z + 0

Theorem 7. (see [31]). Let z ∈ D(A) and F ∈ C(I , X). If F ∈ L1 (I , [D(A)]) then the mild solution of (8)-(9) is a classical solution. Theorem 8. (see [31]). Let z ∈ D(A) and F ∈ C(I , X). If F ∈ W1,1 (I , X) then the mild solution of (8)-(9) is a classical solution.


375

In the subsequent result, we signify by (−A)ϑ the fractional power of the operator −A, (see [7] for details). Lemma 9. (see [28]). Suppose that the conditions P1-P3 are satisfied. Let α ∈ (1, 2) and ϑ ∈ (0, 1) such that αϑ ∈ (0, 1) then there exists a positive number C such that (−A)ϑ Rα (t) ≤ Cert t −αϑ , ϑ

(11)

rt α (1−ϑ )−1

(−A) Sα (t) ≤ Ce t

,

(12)

for all t > 0. 3 Existence results Definition 5. A function x : (−∞, ´ s T) → X, is called a mild solution of (1)-(2) on [0, T], if x0 = ϕ ; x|[0,T] ∈ C([0, T] : X); the function s → 0 Sα (t − s)F (s − τ )g(τ , Xρ (τ ,xτ ) )d τ is integrable on [0,t) for all t ∈ (0, T], ˆ tˆ s x(t) = Tα (t)[ ϕ (0) − g(0, ϕ (0))] + g(t, xρ (t,xt ) ) − Sα (t − s)F (s − τ )g(τ , Xρ (τ ,xτ ) )d τ ds 0 0 ˆ s ˆ t Sα (t − s)H (s, xρ (s,xs ) , k(s, τ , xρ (τ ,xτ ) ))d τ )ds. + 0

0

Here, we initializing the suppositions as H1 The family of operators Tα (t) and Sα (t) are compact for all t > 0, and there exists some constants M, M1 provided that Tα (t) L (X) ≤ M and Sα (t) L (X) ≤ M1 for every t ∈ I , moreover (−A)ϑ Sα (t) L (X) ≤ Mt α (1−ϑ )−1,

0 ≤ t ≤ T.

H2 The following conditions are satisfied. a) F (·)x ∈ C([0, T], X) for every x ∈ [D((−A)1−ϑ )], b) There is a function μ (·) ∈ L1 (I , R+ ), such that F (s)Sα (t) L ([D((−A)ϑ )], X) ≤ M μ (s)t αϑ −1 ,

0 ≤ s ≤ t ≤ T.

H3 The function g(·) is (−A)ϑ −valued, g : I × Bh → [D((−A)−ϑ )] is continuous and there exists positive constant Lg > 0 and Lg∗ > 0, to ensure that (−A)ϑ g(t, E1 ) − (−A)ϑ g(t, E2 )X ≤ Lg (E1 − E2 Bh ),

t ∈I,

E1 , E2 ∈ Bh

ϑ

and (−A) g(t, E )X ≤ Lg (E + 1)Bh where Lg∗ = max(−A)ϑ g(t, 0)X . H4 k : D × Bh → X is continuous and some constants Lm > 0 and Lm∗ > 0, such that k(t, s, ς ) − k(t, s, η ) X ≤ Lm ς − η Bh

and Lm∗ = max k(t, s, 0) X . t∈I

H5 The function H (·) is (−A)ϑ - valued H : I × Bh × X → X is continuous and we can find positive constants L f > 0 and L f∗ > 0, in such a way that, for t ∈ I , H (t, ς1 , x) − H (t, ς2 , y)X ≤ L f ς1 − ς2 Bh + L f x − yX and L f = max H (t, 0, 0)X . t∈I


376

H6 The following inequalities hold. i) For some m > 0 ˆ M Tαϑ t μ (τ )d τ ](Lg (k1∗ r + kn∗ ) + M1 Lg∗ ) αϑ 0 +MT[(k1∗ r + kn )(L f + L f TLm ) + L f Lm∗ T + L f∗ ] ≤ m,

MM1 [Lg (ϕ Bh + Lg∗ )] + [M1 +

ii) Assume n=

k1∗ [MT(L f

M Tαϑ + TL f Lm )] + [M1 + αϑ

ˆ

t 0

μ (τ )d τ ](Lg )x − yB0 ,

where 0 < n < 1.

T

Theorem 10. Assume that the conditions H1-H6 hold. Then the system (1)-(2) has a unique solution on [0, T]. Proof. Now we transforming the system (1)-(2) into a fixed point problem. Let us define the operator Π : BT → BT by (Πx)(t) = Tα (t)[ ϕ (0) − g(0, ϕ (0))] + g(t, xρ (t,xt ) ) ˆ tˆ s Sα (t − s)F (s − τ )g(τ , Xρ (τ ,xτ ) )d τ ds − 0 0 ˆ s ˆ t Sα (t − s)H (s, xρ (s,xs ) , k(s, τ , xρ (τ ,xτ ) ))d τ )ds, + 0

t ∈I.

(13)

0

To ensure the fixed points of the operator Π are the mild solution of the system (1)-(2), we express the function p(·) : (−∞, T] → X by ϕ (t), t ≤ 0; p(t) = t ∈I, Tα (t)ϕ (0), then p0 = ϕ . For every function q ∈ C(I , R) together q(0) = 0, in a way that 0, t ≤ 0; q(t) = q(t), t ∈I. If the function x(·) satisfies (13) then we can introduce x(t) as x(t) = p(t) + q(t), t ∈ I . Moreover, the function q(·) satisfies q(t) = Tα (t)g(0, ϕ (0)) + g(t, pρ (t,pt +qt ) + qρ (t,pt +qt ) ) ˆ tˆ s Sα (t − s)F (s − τ )g(τ , pρ (τ ,pτ +qτ ) + qρ (τ ,pτ +qτ ) )d τ ds − 0 0 ˆ s ˆ t Sα (t − s)H (s, pρ (s,ps +qs ) + qρ (s,ps +qs ) , k(s, τ , pρ (τ ,pτ +qτ ) + qρ (τ ,pτ +qτ ) ))d τ )ds. + 0

0

Let us define BT0 = {q ∈ BT0 : q0 = 0 ∈ Bh } and the semi norm in BT0 as qB0 = sup z(t)X , q ∈ BT0 . T

We construct the operator Π :

BT0

→

BT0

by

(Πq)(t) = Tα (t)g(0, ϕ (0)) + g(t, pρ (t,pt +qt ) + qρ (t,pt +qt ) ) ˆ tˆ s Sα (t − s)F (s − τ )g(τ , pρ (τ ,pτ +qτ ) + qρ (τ ,pτ +qτ ) )d τ ds − 0

0

t∈I


ˆ + 0

t

ˆ Sα (t − s)H (s, pρ (s,ps +qs ) + qρ (s,ps +qs ) ,

s 0

377

k(s, τ , pρ (τ ,pτ +qτ ) + qρ (τ ,pτ +qτ ) )d τ )ds.

It is evident that the operator Π has a fixed point if and only if Π has a fixed point. In this connection we consider Bl = {x ∈ X : x ≤ l} for some l > 0. Now (Πq)(t)X ≤ I1 + I2 + I3 + I4 . Where I1 = Tα (t)X g(0, ϕ (0))X ≤ MM1 [Lg (ϕ Bh + Lg∗ )]. I2 = g(t, pρ (t,pt +qt ) + qρ (t,pt +qt ) )X ≤ (−A)−ϑ [(−A)ϑ g(t, pρ (t,pt +qt ) + qρ (t,pt +qt ) ) − (−A)ϑ g(t, 0)X + (−A)ϑ g(t, 0)X ]

≤ M1 [Lg pρ (t,pt +qt ) + qρ (t,pt +qt ) Bh + Lg∗ ]

≤ M1 [Lg {pρ (t,pt +qt ) Bh + qρ (t,pt +qt ) Bh } + Lg∗ ]

≤ M1 [Lg {k1∗ sup q(t)X + (k1∗ μ H + k2∗ + J ς ))φ Bh } + Lg∗ ] ≤

0≤T≤s ∗ M1 Lg (k1 r + kn∗ ) + M1 Lg∗ . ˆ tˆ s

I3 = Sα (t − s)F (s − τ )g(τ , pρ (τ ,pτ +qτ ) + qρ (τ ,pτ +qτ ) )d τ dsX ˆ t0ˆ s0 ≤ μ (s − τ )M(t − s)αϑ −1[(−A)ϑ g(τ , pρ (τ ,pτ +qτ ) + qρ (τ ,pτ +qτ ) ) − (−A)ϑ g(τ , 0)X 0

0

+(−A)ϑ g(τ , 0)X ]d τ ds ˆ M Tαϑ t μ (τ )d τ )[Lg (k1∗ r + kn∗ ) + Lg∗ ]. ≤ ( αϑ 0 ˆ t ˆ s Sα (t − s)H (s, pρ (s,ps +qs ) + qρ (s,ps +qs ) , k(s, τ , pρ (τ ,pτ +qτ ) + qρ (τ ,pτ +qτ ) )d τ )dsX I4 = ˆ0 t ˆ0 s ≤ [ Sα (t − s)H (s, pρ (s,ps +qs ) + qρ (s,ps +qs ) , k(s, τ , pρ (τ ,pτ +qτ ) + qρ (τ ,pτ +qτ ) )d τ )ds 0

0

−H (s, 0, 0)X + H (s, 0, 0)X ]ds. ˆ s ˆ t (k(s, τ , pρ (τ ,pτ +qτ ) + qρ (τ ,pτ +qτ ) ) ≤ M[ L f pρ (s,ps +qs ) + qρ (s,ps +qs ) Bh + L f 0

−k(s, τ , 0) + k(s, τ , 0)X )d τ + L f∗ ]ds

0

≤ MT[(k1∗ r + kn )(L f + L f TLm ) + L f Lm∗ T + L f∗ ] Therefore (Πq)(t) ≤ MM1 [Lg (ϕ Bh + Lg∗ )] + M1 [Lg (k1∗ r + kn∗ ) + Lg∗ ] ˆ M Tαϑ t μ (τ )d τ )[Lg (k1∗ r + kn∗ ) + M1 Lg∗ ] +( αϑ 0 +MT[(k1∗ r + kn )(L f + L f TLm ) + L f Lm∗ T + L f∗ ] ˆ M Tαϑ t ∗ μ (τ )d τ ](Lg (k1∗ r + kn∗ ) + M1 Lg∗ ) ≤ MM1 [Lg (ϕ Bh + Lg ) + [M1 + αϑ 0 +MT[(k1∗ r + kn )(L f + L f TLk ) + L f Lk∗ T + L f∗ ] ≤ m. Therefore, we estimates that Π maps the ball Bl (0, BT0 ) into itself. Finally, we show that Π is a

378


contraction on Bl (0, BT0 ). So that consider x, y ∈ Bl (0, BT0 ), then (Πx)(t) − (Πy)(t)X

M Tαϑ ≤ + TL f Lm )] + [M1 + αϑ ≤ nx − yB0 . k1∗ [MT(L f

ˆ 0

t

μ (τ )d τ ](Lg )x − yB0

T

T

From the hypothesis H6 and the reference to the contraction mapping principle, we ensure that Π has a unique fixed point q ∈ BT0 which is the mild solution of the system (1)-(2). This completes the proof. 4 An example To demonstrate our theoretical outcomes, we treat the FNIDS with SDD of the model ˆ 1 t 6(s−t) Dtα [x(t, ξ ) − e x(s − ρ1 (s)ρ2 (x(s)), ξ )ds] 36 −∞ ˆ 1 t 6(s−t) ∂2 x(t, ξ ) − e x(s − ρ1 (s)ρ2 (x(s)), ξ )ds = ∂ξ2 36 −∞ ˆ t ∂2 + (t − s)δ e−λ (t−s) 2 x(t, ξ )ds ∂ξ 0 ˆ t 1 e6(s−t) x(s − ρ1 (s)ρ2 (x(s)), ξ )ds + 25 −∞ ˆ ˆ t 1 t sin(t − s) e6(s−t) x(s − ρ1 (s)ρ2 (x(s)), ξ )d τ ds + 16 0 −∞ x(t, 0) = x(t, π ), t ∈ [0, T], x(t, ξ ) = ϕ (t, ξ ), t ≤ 0, ξ ∈ [0, π ],

(14) (15) (16)

where Dtα is Caputo’s fractional derivative of order α ∈ (1, 2), η and λ are positive numbers and ϕ ∈ Bh . To exhibit this structure (14)-(16) in the abstract form (1)-(2), we consider the space X = L2 ([0, π ]). In the sequel, A : D(A) ⊂ X → X by Ax = x with domain D = {x ∈ X : x, x are absolutely continuous, x ∈ X, x(0) = x(π ) = 0}. Then ∞

Ax =

∑ n2 x, xn xn ,

x ∈ D(A),

n=1

such that xn (s) = π2 sin(ns), n = 1, 2, .... is the orthogonal set of eigenvectors of A. The infinitesimal generator A of an analytic semigroup T (t)t≥0 in X is characterized by ∞

T (t)x =

∑ e−n x, xn xn , 2

x ∈ X,

t > 0.

n=1

Let us take δ = 1/2 then (−A)1/2 is defined by (−A)1/2 =

∞

∑ n x, xn xn ,

x ∈ (D(−A)1/2 ) and (−A)−1/2 = 1.

n=1

´0 For the phase space, choose h = e6s , s < 0, then d = −∞ h(s)ds = 1/6 < ∞, for t ≤ 0 and determine ˆ 0 h(s) sup ϕ (θ )L2 ds. ϕ = −∞

θ ∈[s,0]


379

Therefore, for (t, ϕ ) ∈ [0, a] × Bh , where ξ (θ )(ξ ) = ξ (θ , ξ ), (θ , ξ ) ∈ (−∞, 0] × [0, π ]. Set x(t)(ϕ ) = x(t, ϕ ),

ρ (t, ϕ ) = ρ1 (t)ρ2 (ϕ (0)),

then we can have ˆ

ϕ ds, 36 −∞ ˆ 0 ϕ e6(s) ds + (K ϕ )(ξ ), H (t, ϕ , K ϕ )(ξ ) = 25 −∞

g(t, ϕ )(ξ ) =

0

e6(s)

where ˆ (K ϕ )(ξ ) =

t

ˆ sin(t − s)

0 −∞

0

e6(s)

ξ d τ ds. 16

In addition, we can prove H (t, ϕ , K ϕ )L2

ˆ π ˆ ≤ ( ( 0

0

−∞

6(s)

e

ϕ ds + 25

ˆ 0

t

ˆ sin(t − s)

0 −∞

e6(τ )

ξ d τ ds)2 d ξ )1/2 16

ˆ 0 ˆ ˆ π 1 0 6(s) 1 6(s) ( ≤ ( e sup ϕ ds + e sup ϕ ds)2 d ξ )1/2 16 −∞ 0 25 −∞ √ √ π π ϕ Bh + ϕ Bh ≤ 25 16 ≤ LH ϕ Bh + L H ϕ Bh , where LH + L H =

√ 41 π 400 ,

and

H (t, ϕ , K ϕ ) − H (t, ϕ, K ϕ)L2 ˆ t ˆ π ˆ 0 ˆ 0 ϕ ϕ ϕ ϕ ≤( ( e6(s) − ds + sin(t − s) e6(τ ) − d τ ds)2 d ξ )1/2 25 25 16 16 −∞ 0 −∞ 0 ˆ 0 ˆ 0 ˆ π 1 1 ds + ( ≤( e6(s) sup ϕ − ϕ e6(s) sup ϕ − ϕds)2 d ξ )1/2 25 16 −∞ −∞ 0 √ √ π π ϕ − ϕBh + ϕ − ϕBh ≤ 25 16 ≤ LH ϕ − ϕBh + L H ϕ − ϕBh .

Along these modifications, the model (14)-(16) is transformed into the model (1)-(2). If all the assumptions H1-H6 are fulfilled, then from Banach contraction principle, the system (14)-(16) has a mild solution on (−∞, 0].

5 Conclusion This article deals with the study of ”Existence result for a neutral fractional integro-differential equation with state dependent delay in Banach spaces”. We have done an investigation on neutral fractional integro differential equation with SDD by utilizing Banach contraction principle with resolvent operator technique. To validate our theoretical results, we analyzed an example. In future we will extend the present study to establish the other quantitative and qualitative aspects.

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[8] Podlubny, I. (1999), Fractional Differential Equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego. [9] Zhou, Y. (2014), Basic Theory of Fractional Differential Equations, World Scientific, Singapore. [10] Agarwal, R.P., Lupulescu, V., O’Regan, D. and Rahman, G. (2015), Fractional calculus and fractional differential equations in nonreflexive Banach spaces, Commun. Nonlinear. Sci. Numer. Simul., 20(1), 59-73. [11] Byszewski, L. (1991), Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal cauchy problem, J. Math. Anal. Appl., 162, 494-505. [12] Banano, G.R., and Rodriguez-Lopez, S. (2014), Tersian Existence of solutions to boundary value problem for impulsive fractional differential equations, Frac. calc. Appl.Anal., 17(3), 717-744. [13] Chang, Y.K. and Nieto, J.J. (2009), Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators, Numer. Funct. Anal. Optim., 30(3-4), 227-244. [14] Lv, Z. and Chen, B. (2014), Existence and uniquenes of positive solutions for a fractional swithced system, Abstr. Appl. Anal., 7, 2014, Article ID 828721. [15] Ravichandran, C. and Baleanu, D. (2013), Existence results for fractional neutral functional integrodifferential evolution equations with infinite delay in Banach spaces, J. Adva. Diff. Equ., 1, 215-227. [16] Suganya, S., Mallika Arjunan, M., and Trujillo, J.J. (2015), Existence results for an impulsive fractional integro-differential equation with state-dependent delay, App. Math. comput., 266, 54-69. [17] Valliammal, N., Ravichandran, C., and Park, J.H. (2017), On the controllability of fractional neutral integrodifferential delay equations with nonlocal conditions, Math. Methods Appl. Sci., 40, 5044-5055. [18] Vijayakumar, V., Ravichandran, C., and Murugesu, R. (2014), Existence of mild solutions for nonlocal cauchy problem for fractional neutral evolution equations with infinite delay, Surv. Math. Appl., 9, 117-129. [19] Vijayakumar, V., Ravichandran, C., and Murugesu, R. (2013), Approximate controllability for a class of fractional neutral integro-differential inclusions with state-dependent delay, Nonlinear Stud., 20(4), 511-530. [20] Wang, Y., Liu, L., and Wu, Y. (2014), Positive solutions for a class of higher order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters, Adv. Difference. Equ., 268, 1-24. [21] Zhou, Y., Zhang, L., and Shen, X.H. (2014), Existence of mild solutions for fractional evolution equations, J. Integral Equations Appl., 25, 557-586. [22] Zhou, Y. and Jiao, F. (2010), Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59, 1063-1077. 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Synchronization of Unified Chaotic System via Output Feedback Control Scheme Xue-Rong Tao1 , Ling Tang2 , Ping He3† 1

2

3

School of Physics & Electronic Engineering, Sichuan University of Science & Engineering, Zigong, Sichuan, 643000, People’s Republic of China School of Automation & Information Engineering, Sichuan University of Science & Engineering, Zigong, Sichuan, 643000, People’s Republic of China Emerging Technologies Institute, The University of Hong Kong, Pokfulam, Hong Kong Submission Info

Abstract

Keywords

This article presents a new method of synchronization of unified chaotic system by employing output feedback control strategy. In particular, for unified chaotic system with parameter α ∈ [0, 1], we design explicit and simple output feedback control scheme by which the equilibrium point of error system is globally stabilized. The numerical simulation of unified chaotic system has been demonstrated to show the effectiveness of the proposed synchronization scheme.

Synchronization Unified chaotic system Output feedback control

©2018 L&H Scientific Publishing, LLC. All rights reserved.

Communicated by A.C.J. Luo Received 21 March 2017 Accepted 2 September 2017 Available online 1 January 2019

1 Introduction Chaotic system is a complex dynamical nonlinear system and its response exhibits some specific characteristics [1] such as excessive sensitivity to initial conditions [2], broad Fourier transform spectrums [3], and irregular identities of the motion in phase space [4]. Also, it has been found to be useful in analyzing many problems, such as information processing [5], power systems collapse prevention [6], high-performance circuits and devices [7], etc. Since the synchronization of chaotic system has been proposed by Pecora and Carroll in 1990 [8], a wide variety of approaches have been proposed for synchronization of chaotic systems such as generalized synchronization [9], phase synchronization [10], projective synchronization [11], output feedback H ∞ synchronization [12], lag-synchronization [13], adaptive synchronization [14], etc. Most of the existing theoretical results mainly focus on the systems whose models are identical or similar, and parameters are exactly known in advance. Controlling a unified chaotic system by using output feedback control strategy was presents [15]. To the knowledge of the authors, no study on the synchronization of unified chaotic system has † Corresponding



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been carried out with the output feedback control scheme. In this paper, further investigation on the synchronization of the unified chaotic system is explored. The rest of this paper is organized as follows. In Section 2, the related unified chaotic system is described. In Section 3, the output feedback control scheme has been employed for achieving the synchronization of the unified chaotic system. In Section 4, numerical simulations have been presented in figures to confirm the effectiveness of the output feedback control scheme. Finally, conclusions are given in Section 5. 2 System analysis According to the structure of Lorenz system [16], L¨ u system [17], and Chen systerm [18], the unified chaotic system was proposed by L¨ u [19], which is described by  x˙1 =(25α + 10)(x2 − x1 ),    x˙2 =(28 − 35α )x1 + (29α − 1)x2 − x1 x3 ,   8+α  x3 . x˙3 =x1 x2 − 3

(1)

where x = [x1 , x2 , x3 ]T ∈ R3 is the state variables of unified chaotic system (1) and α ∈ [0, 1] is a system parameter. when α ∈ [0, 0.8), the system (1) is Lorenz chaotic system, when α = 0.8, the system (1) is L¨ u chaotic system, when α ∈ (0.8, 1], the system (1) is Chen chaotic system. So, the system (1) is regarded as unified chaotic system. In particular, the system (1) bridges the gap between Lorenz system and Chen system. With initial condition x(0) = [1, 0, −1]T , the unified chaotic system have some chaotic attractors, which are shown in Figures 1-3. The state responses are shown in Figures 4-6. We take system (1) as the drive system, and construct the output as y = x2 .

(2)

Xue-Rong Tao et al. / Journal of Applied Nonlinear Dynamics 7(4) (2018) 383–392

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We construct the controlled response system as follows:  x˙ˆ1 =(25α + 10)(xˆ2 − xˆ1 ) + u1 ,       x˙ˆ2 =(28 − 35α )xˆ1 + (29α − 1)xˆ2 − xˆ1 xˆ3 + u2 , 8+α  xˆ3 , xˆ˙3 =xˆ1 xˆ2 −   3    yˆ =xˆ2 .

(3)

where xˆ = [xˆ1 , xˆ2 , xˆ3 ]T ∈ R3 is the state of response system (3), y is the output of response system (3), u

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is controller satisfying ui (0) = 0, (i = λ ). Let error dynamics e = xˆ − x.

(4)

The error dynamical system can be obtained as  e˙1 =(25α + 10)(e2 − e1 ) + u1 ,       e˙2 =(28 − 35α )e1 + (29α − 1)e2 − e1 e3 − e1 x3 − x1 e3 + u2 , 8+α  e˙3 =e1 e2 + e1 x2 + e2 x1 − e3 ,   3    ey =e2 .

(5)

3 Main result In this section, we will study the synchronization problem of the drive system (1) and the response system (3). Theorem 1. The following controller u1 = k1 ey , k1 = −(10 + 25α ),

(6-1)

u2 = k2 ey , k2 = 1 − 29α − µ , (µ > 0).

(6-2)

can be used to synchronize the drive system (1) and the response system (3).

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Proof. Substituting the controller (6-1) and (6-2) into the error dynamical system (5), we have e˙1 =(25α + 10)(e2 − e1 ) − k1 e2 = − (25α + 10)e1 , e˙2 =(28 − 35α )e1 + (29α − 1)e2 − e1 e3 − e1 x3 − x1 e3 + k2 e2 ,

(7-1) (7-2)

8+α e3 . (7-3) 3 According to the structure of solution of first-order linear differential equation and (7-1), we have e˙3 = e1 e2 + e1 x2 + e2 x1 −

e1 (t) = e1 (0)e−(25α +10)t .

(8)

lim e1 (t) = 0.

(9)

Because of α ∈ [0, 1], that is, yield t→+∞

which implies that e1 (t) is exponentially stable. Substituting the (9) into (7-2) and (7-3), thus, the error dynamical system (7-1), (7-2) and (7-3) can be reduced as follows: e˙2 =(29α − 1)e2 − x1 e3 + k2 e2 = − x1 e3 − µ e2 , 8+α e3 . 3 Next, we can construct positive definite quadratic Lyapunor function. as follows: e˙3 = e2 x1 −

V (t) = e22 + e23 .

(10-1) (10-2)

(11)

Differentiation V (t) with respect to time along the solution of system (10-1) and (10-2) results into: dV (t) =2e2 e˙2 + 2e3 e˙3 dt

8+α e3 ] 3 2(8 + α ) 2 =2(29α − 1)e22 − 2x1 e2 e3 + 2k2 e22 + 2x1 e2 e3 − e3 3 2(8 + α ) 2 e3 . =2(29α − 1 + k2 )e22 − 3 Substituting the controller (6-2) into the (12), yied dV (t) 2(8 + α ) 2 ≤ 0, ∀e2 , e3 , 2 = −2µ e2 − e3 6= 0, ∀ k [e2 , e3 ]T k6= 0. dt 3 =2e2 [(29α − 1)e2 − x1 e3 + k2 e2 ] + 2e3 [e2 x1 −

Then, according to Lyapunov stability theory, we have   lim e2 (t) = 0, t→+∞

 lim e3 (t) = 0.

(12)

(13)

(14)

t→+∞

According to (9) and (14), yield

lim e(t) = 0.

t→+∞

(15)

Thus, the drive system (1) and the response system (3) with controller (6-1) and (6-2) can be synchronize. The proof is completes.


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4 Numerical simulation We will show a series of numerical experiments by using the fourth-order Runge-Kutta method with step size 0.001. The initial conditions of drive system (1) and response system (3) are as following:         xˆ1 (0) −4.2 8.0 x1 (0)  x2 (0)  =  3.0  ,  xˆ2 (0)  =  −5.6  . xˆ3 (0) 3.4 −10.0 x3 (0) Let the parameter µ of controller (6-1) and (6-2) as follows:

µ = 10. The synchronization errors with different α and different µ between the drive system (1) and response system (3) are shown in Figures 7-9. They are shown that the synchronization errors e(t) converge to zero when the controller has been activated. The curves of the (6-1) and (6-2) are shown in Figures 10-11. 5 Conclusion In this paper, output feedback controllers are presented to synchronize a unified chaotic system when the states are not all mensurable. Compared with the present results, the controllers designed in this paper have many advantages, such as small feedback gain, simple structure and less conservation. Acknowledgment This work was jointly supported by National Natural Science Foundation of China (Grant Nos. 11705122, 61640223), Research Foundation of Department of Education of Sichuan Province (Grant no. 17ZA0271),

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Fig. 9 State response of synchronization errors e1 (t), e2 (t) and e3 (t) with α = 1 and µ = 30.

Open Foundation of Enterprise Informatization and Internet of Things Key Laboratory of Sichuan Province (Grant no. 2016WYJ03), Open Foundation of Artificial Intelligence Key Laboratory of Sichuan Province (Grant no. 2016RYJ04), and Open Foundation of Sichuan Provincial Key Lab of Process Equipment and Control (Grant no. GK201612).


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References [1] He, P. (2016), Chaos Control of Chaotic Systems: Stabilization, Synchronization & Circuit Implementation, LAP LAMBERT Academic Publishing, Saarbr¨ ucken, Germany, 1th edition. [2] Nik, H.S., He, P., and Talebian, S.T. (2014), Optimal, adaptive and single state feedback control for a 3D chaotic system with golden proportion equilibria, Kybernetika, 50(4), 596-615. [3] Chen, C.Z., He, P., Fan, T., and Jing, C.G. (2015) Finite-time chaotic control of unified hyperchaotic systems with multiple parameters, International Journal of Control and Automation, 8(8), 57-66. [4] He, P. and Fan, T. (2016), Delay-independent stabilization of nonlinear systems with multiple time-delays and its application in chaos synchronization of Rössler system. International Journal of Intelligent Computing and Cybernetics, 9(2), 205-216. [5] He, P. and Li, Y.M. (2015), Control and synchronization of a hyperchaotic finance system via single controller scheme, International Journal of Intelligent Computing and Cybernetics, 8(4), 330-334. [6] Jing, C.G., He, P., Fan, T., Li, Y.M, Chen, C.Z., and Song, X. (2015), Single state feedback stabilization of unified chaotic systems and circuit implementation, Open Physics, 13(1), 111-122.

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A Novel Quasigroup Substitution Scheme for Chaos Based Image Encryption Vinod Patidar†, N. K. Pareek, G. Purohit Department of Physics, Sir Padampat Singhania University, Bhatewar, Udaipur 313601, Rajasthan, India Submission Info Communicated by A.C.J. Luo Received 1 July 2017 Accepted 12 September 2017 Available online 1 January 2019 Keywords Image encryption Quasigroup Chaos Latin squares Permutation Substitution

Abstract During last two decades, there has been prolific growth in the chaos based image encryption algorithms. Up to an extent these algorithms have been able to provide an alternative to exchange large media files (images and videos) over the networks in a secure way. However, there have been some issues with the implementation of chaos based image ciphers in practice. One of them is reduced/small key space due to the fact that chaotic behavior is only observed for certain range of system parameters/initial conditions of the chaotic system used in such algorithms. To overcome this difficulty, we propose a simple, efficient and robust image encryption algorithm based on combined applications of quasigroups and chaotic standard map. The proposed image cipher is based on the Shannon’s popular substitution-diffusion architecture where a quasigroup of order 256 and chaotic standard map have been used for the substitution and permutation of image pixels respectively. Due to the introduction of quasigroup as part of the secret key along with the parameter and initial conditions of the chaotic standard map, the key space has been increased significantly. The proposed image cipher is very fast due to the fact that the substitution based on the quasigroup operations is very simple and can be executed easily through the lookup table operations on Latin squares (which are Cayley operation tables of quasigroups) and the permutation is performed row-by-row as well as column-by-column using the pseudo random number sequences generated through the chaotic standard map. The security and performance have been analyzed through the histograms, correlation coefficients, information entropy, key sensitivity analysis, differential analysis, key space analysis etc. and the results prove the efficiency and robustness of the proposed image cipher against the possible security threats. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Chaos based image encryption has been an active area of research over the last few years which has proven that a suitable choice of chaotic systems along with an intelligently designed combination of † Corresponding



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mixing transformations (substitution and diffusion transformations) comprising of complex involvement of the key leads to a very fast and potentially reliable image encryption algorithm [1-8]. The dynamics of the chaotic system, used in most of the chaos based cryptosystems, is control/system parameter dependent therefore they exhibit desired chaotic behavior only for certain ranges/values of control/system parameters. Many of the chaos based cryptosystems utilize system parameter / initial conditions of the chaotic systems as part of the secret key hence such cryptosystems mainly suffer from key size requirements [9-13]. Despite this drawback of having a reduced key space, the chaotic systems have been a preferred choice [14-18] for the image encryption systems due to their inherent features: ergodicity, mixing property, sensitivity to initial conditions/system parameters, etc. which closely resemble with the ideal cryptographic properties: confusion, diffusion, balance, avalanche properties, etc. [9]. On the other hand conventional number theory based cryptographic algorithms [19] utilize associative algebraic structures such as groups, rings and fields as the main ingredient to achieve desired level of confusion and diffuision. Contemporaneous to the development of chaos based cryptography, Dénes and Keedwell [20, 21] proclaimed the start of a new era in cryptography by using the non-associative algebraic structures such as quasigroups and neo-fields. Since then there have been some attempts of designing quasigroup based pseudorandom number generators/cryptographic schemes [22-29] but some weaknesses have been found in them [30] and have seen relatively less success than chaos based systems. Some of the comprehensive and in-depth overviews of the recent developments and current state of the art in the field of applications of quasigrous in cryptology can be seen in [31, 32]. Partly the interest in utilizing quasigrous in cryptography raised due to the following reasons: (i) the algebraic structures known as Quasigroups [33] provide an alternate and powerful method for generating a larger set of permutation/substitution transformations by rearranging not only the data values, but also changing the data values across their range, (ii) the Cayley table (the binary operation table) of a finite quasi group of order n is always a Latin square [34] and the number of Latin squares of order n increases very quickly with n and indeed very large for rather small n [35, 36]. Hence, such transformations can provide a potent combination of confusion and diffusion in the encryption of digital images and also the quasigroup based cryptosystem can generate more number of keys even we consider a small order quasiqroup. As described above that the chaos based cryptosystems mainly suffer from key size requirements and the quasigroups can provide a larger set of permutation/substitution transformations i.e., an immensely large number of keys. Keeping these facts in mind we propose in this paper a simple, efficient and robust image encryption algorithm based on combined applications of quasigroups and chaotic dynamical systems. The proposed image cipher is a block cipher and based on the Shannon’s substitution-diffusion architecture [37] which has been the main core structure of most of the chaos based image encryption algorithms developed during last two decades. In the proposed image cipher we use a quasigroup of order 256 and chaotic standard map for the substitution and permutation of image pixels respectively. The secret key of the image cipher comprises of a quasigroup of order 256, initial conditions & parameter of chaotic standard map. The encryption in the proposed image cipher has minimum (maximum) two (sixteen) rounds and each round has a substitution and a permutation module. The substitution is done column-wise and row-wise starting from the first and last pixel respectively through the left and right quasigroup binary operations on a 256 order quasigroup (which is part of the secret key). However the permutation is performed row-by-row as well as column-by-column using the pseudo random number sequences generated through the chaotic standard map. Due to introduction of quasigroup as the part of secret key, the proposed image cipher does not suffer with the key size requirement (as opposed to the most of the existing chaos based image ciphers) since the no. of possible quasigroups of order 256 lies somewhere in the range 0.753 × 10102805 ≥ L(256) ≥ 0.304 × 10101724 [35, 36]. The proposed image cipher is very fast due to the fact that the substitution based on the quasigroup operations are simple and can be executed easily through the lookup table operations on Latin squares (which are Cayley operation


⊛ 1 2 3 4 5

1 2 5 3 4 1

2 1 4 5 2 3

3 5 2 1 3 4

4 3 1 4 5 2

5 4 3 2 1 5

W 1 2 3 4 5

1 2 4 3 5 1

2 1 3 5 2 4

3 4 5 1 3 2

4 5 2 4 1 3

5 3 1 2 4 5

⊘ 1 2 3 4 5

1 5 1 3 4 2

2 1 4 5 2 3

3 3 2 4 5 1

4 2 5 1 3 4

395

5 4 3 2 1 5

Fig. 1 Quasigroup of order 5 and its left and right inverses

tables of quasigroups) without any explicit binary quasigroup operation calculation. We also present the results of our security and performance analysis of the proposed image encryption done through the histograms, correlation coefficients, information entropy, key sensitivity analysis, differential analysis, key space analysis etc. Results are encouraging and illustrate the efficiency and robustness of the proposed image cipher against the possible security threats. The rest of the paper is organized as follows: In Section 2, we briefly introduce the notion of quasigroup and associated operations which have been used in the proposed algorithm. In Section 3 we describe in detail the complete process of encryption and decryption used in the proposed image cipher. In Sections 4 to 7, we summarize the results of our statistical, key-sensitivity, differential and key space analyses to assess the performance and robustness of the proposed image cipher. Finally Section 8 concludes the paper. 2 The quasigroup The structure (Q, ⊛), Q = {q1 , q2 , q3 , q4 , . . . , qn }, Q = n is called a finite quasigroup of order n [33] if, ∀a, b ∈ Q, ∃x, y ∈ Q such that a ⊛ x = b and y ⊛ a = b and ..... ..... ∀x, y, z ∈ Q, x ⊛ y = x ⊛ z ⇒ y = z and y ⊛ x = z ⊛ x ⇒ y = z. Thus the Cayley table of a finite quasigroup of order n is a Latin square of order n and vice-versa. A Latin square is an n × n array filled with n different symbols in such a way that each of the symbol occurring exactly once in each row and exactly once in each column [34]. With every quasigroup (Q, ⊛) there exist two more operations W and ⊘ satisfying the following ∀x, y, z ∈ Q such that x ⊛ y = z ⇔ y = x W z ⇔ x = z ⊘ y. ¨¨˙

¨¨˙

These operations W and ⊘ can be referred as left and right inverse quasigroup operations respectively and the structures (Q, W) and (Q, ⊘) also form quasigroups of the same order. In Fig. 1, we have given example of an order 5 quasigroup along with its left and right inverse. The first table in the Figure 1 is the Caley operation table (Latin square) of an order 5 quasigroup, which is any random arrangement of numbers 1 to 5 in a 5X5 grid in such a way that each number appears only once in each row and exactly once in each column. There are many such arrangements possible, in fact for this case of order 5 quasigroup, number of such possible arrangements is 161280. The second table in Figure 1 is the Caley operation table for the left inverse quasigroup. For generating left inverse quasigroup operation table, we may use the following look up rule: for finding result of x W y just look for the column number of element ‘y’ in the xth row of the original quasigroup operation table. Similarly the right inverse quasigroup operation table can be obtained using the following look up rule: for finding result of x ⊘ y just look for the row number of element ‘x’ in the yth column of the original quasigroup operation table. The result of right inverse quasigroup operation for the quasigroup of order 5 (shown in the first table of Figure 1) is shown in the third table of Figure 1. The number of Latin squares of order n increases very quickly with n and indeed very large for rather small n. The total number of all possible Latin squares L(n) of order n × n is only known up to n = 11,

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which is L(11) = 776966836171770144107444346734230682311065600000. The most accurate upper and lower bound for the number of Latin squares of order n is [32, 33] given by following expression n Y

(k!)n/k ≥ L(n) ≥

k=1

(n!)2n . nn2

(1)

According to this, the upper and lower bounds for the L(256) are: 0.753 × 10102805 ≥ L(256) ≥ 0.304 × 10101724 . Therefore we may safely conclude that there exist at least n! (n − 1)! . . . 2!1! Latin squares of order n. Hence such Latin squares can be potentially used as secret keys in symmetric cryptography. 3 The proposed image cipher 3.1

The original image to be encrypted

The original colour image of height H and width W is the image to be encrypted. The image is read as a 3D matrix PI(i, j, k) of integers between 0 and 255, where 1 ≤ i ≤ H, 1 ≤ j ≤ W and 1 ≤ k ≤ 3. Thus, we have total H × W × 3 integer elements in the 3D input image matrix. 3.2

The secret key

The secret key in the proposed image encryption technique is divided into two parts. First part consists of a set of three floating point numbers and four integers (x0 , y0 , K, NS , NR, seed1, seed2), where (x0 , y0 ) ∈ (0, 2π), K can have any real value greater than 18.0, NS is an integer value preferably greater than 100, NR is an integer value between 1 and 16, which is treated as number of encryption rounds, seed1 and seed2 are integers between 0 to 255 and used as the seed values for the initiation of the substitution. The second part is a 256 × 256 gray image whose gray pixel values (+1) form a 256 × 256 Latin square which is the Cayley operation table of a quasigroup. This table is basically used as the lookup table for the quasigroup substitution. The generation of quasigroups of desired order through software or hardware is very easy [38, 39] hence the secret key of the proposed image encryption scheme can be generated very easily. 3.3

Calculation of the dimension of new 2D matrix

Here we calculate the dimension of 2D matrix (NH and NW) using the values of W and H. It is calculated in such a way that H × W × 3 = NH × NW and (NW − NH) is minimum, hence NW ≥ NH. With the above mentioned conditions, we try to form a square 2D matrix as far as possible from all the elements of 3D image matrix, if it is not possible then we form a rectangular matrix with the lowest possible difference in the number of rows and columns. The H × W × 3 integer elements are then arranged into a 2D matrix P(i, j) where 1 ≤ i ≤ NH and 1 ≤ i ≤ NW. 3.4

Substitution using the quasigroup binary operations

In the proposed image cipher, we use quasigroup operation table to introduce the confusion/substitution. As explained above in the secret key section that we use a quasigroup of order 256 i.e. its Cayley table is a 256 × 256 matrix consisting of integers ranging from 1 to 256 in such a way that no integer occurs twice in a particular row or column. However, the pixel intensity level in RGB images vary from 0 to 255 hence before starting the actual substitution/confusion, we add 1 to all the integers of the 2D matrix P(i, j).


3.4.1

397

Column wise substitution starting from the first pixel (left quasigroup binary operation)

for j = 1 to NW for i = 1 to NH if (i = 1 & j = 1) P(i, j) = seed1 ⊛ P(i, j) else if (i = 1 & j , 1) P(i, j) = P(NH, j − 1) ⊛ P(i, j) else P(i, j) = P(i − 1, j) ⊛ P(i, j) end end end end 3.4.2

Row wise substitution starting from the last pixel (right quasigroup binary operation)

for i = NH to 1 for j = NW to 1 if ( j = NW & i = NH) P(i, j) = P(i, j) ⊛ seed2 else if ( j = NW & i , NH) P(i, j) = P(i, j) ⊛ P(i + 1, 1) else P(i, j) = P(i, j) ⊛ (i, j + 1) end end end end After completion of the quasigroup substitution, subtract 1 from all the integers of the 2D matrix. 3.5

Permutation of rows and columns using chaotic standard map

The permutation of the pixels in the proposed image cipher is a simplified version of the chaotic standard map based permutation used in the pseudorandom-substitution scheme [14]. The execution/algorithmic details of the permutation used in the proposed image cipher are as follows: Iterate the standard map NS times, starting with the initial conditions (x0 , y0 ) and using the parameter K specified in the secret key and the new values are stored as (x, y). Now calculate the number of iteration to skip from the 2D matrix obtained after the substitution in the following way. NS K = {P (1, 1) + P (1, 2) + . . . + P (1, W) + P (2, 1) + . . . + P (NH, NW)} mod 256. Then we generate the permutation boxes and then execute the permutation of rows and columns with the help of chaotic standard map in the following way:

398


for k = 1 to NS K x = (x + K sin y) mod 2π y = (y + x) mod 2π end for j = 1 to NW step 1 x = (x + K sin y) mod 2π y = (y + x) mod 2π x PR1( j) = 1 + ⌊ 2π .NH⌋ y PC1( j) = 1 + ⌊ 2π .NH⌋ x = (x + K sin y) mod 2π y = (y + x) mod 2π x .NH⌋ PR2( j) = 1 + ⌊ 2π y PC2( j) = 1 + ⌊ 2π .NH⌋ end for j = 1 to NH step 1 interchange P(PR1 ( j) , :) and (PR2 ( j) , :) end for j = 1 to NW step 1 interchange P(:, PC1 ( j) , :) and (:, PC2 ( j)) end C (:, :) = P(:, :) In the above algorithm P(i, :), P(:, j) and P(:, :) respectively, represent all the elements of ith row, all the elements of jth column and all the elements of matrix P. These modules of substitution and permutation are repeated sequentially NR number of times (as specified in the secret key). We recommend at least two rounds of encryption to ensure the foolproof security. Finally the elements of 2D matrix C(i, j) are arranged in a 3D matrix CI(i, j, k) of dimension H × W × 3 followed by a conversion to a colour image. It is the final encrypted image. The whole process of encryption is depicted in a simple block diagram in Fig. 2. The process of decryption is completely reverse of the encryption process. For the proper decryption the same secret key is to be supplied and the encrypted image is processed in the same way as in the encryption. The only difference would be that the substitution and permutation operations will be executed in the reverse order. Below we describe the algorithms used for the recovery of permutation and substitution during the decryption process. 3.6

Recover the permutation of rows and columns using chaotic standard map

Iterate the standard map NS times, starting with the initial conditions (x0 , y0 ) and using the parameter K specified in the secret key and the new values are stored as (x, y). NS K = {C (1, 1) + C (1, 2) + . . . + C (1, W) + C (2, 1) + . . . + C (NH, NW)} mod 256. Then we generate the permutation boxes and then execute the permutation of rows and columns with the help of chaotic standard map in the following way: for k = 1 to NS K x = (x + K sin y) mod 2π y = (y + x) mod 2π end for j = 1 to NW step 1 x = (x + K sin y) mod 2π y = (y + x) mod 2π


Fig. 2 Block diagram of the proposed image cipher. x .NH⌋ PR1( j) = 1 + ⌊ 2π y .NH⌋ PC1( j) = 1 + ⌊ 2π x = (x + K sin y) mod 2π y = (y + x) mod 2π x PR2( j) = 1 + ⌊ 2π .NH⌋ y .NH⌋ PC2( j) = 1 + ⌊ 2π

end for j = 1 to NW step −1 interchange C(:, PC2( j), :) and (:, PC1( j)) end for j = 1 to NH step −1 interchange P(PR2( j), :) and (PR1( j), :) end 3.7

Recover the substitution using inverse quasi group operations

Before starting the recovery of substitution add 1 to all the elements of 2D matrix C(i, j). 3.7.1

Recover the row wise substitution (right inverse quasigroup operation)

399

400


CRT (:, :) = C(:, :) for i = NH to 1 for j = NW to 1 if ( j = NW & i = NH) C(i, j) = CRT (i, j) ⊘ seed2 else if ( j = NW & i , NH) C(i, j) = CRT (i, j) ⊘ CRT (i + 1, 1) else C(i, j) = CRT (i, j) ⊘ CRT (i, j + 1) end end end end 3.7.2

Recover the column wise substitution (left inverse quasigroup operation)

CCT (:, :) = C(:, :) for j = 1 to NW for i = 1 to NH if (i =1 & j = 1) C(i, j) = seed1 W CCT (i, j) else if (i = 1 & j , 1) C(i, j) = CCT (NH, j − 1 W CCT (i, j) else C(i, j) = CCT (i − 1, j W CCT (i, j) end end end end P(:, :) = C(:, :) After completion of the recovery of the substitution subtract 1 from all the integers of the output 2D matrix. In the above recovery algorithm ⊘ and W respectively, represent the right inverse and left inverse quasigroup operations. These recovery modules of substitution and permutation are repeated sequentially NR number of times (as specified in the secret key). Finally the elements of 2D matrix P(i, j) are arranged in a 3D matrix PI(i, j, k) of dimension H × W × 3 followed by a conversion to a colour image. It is the decrypted image. 4 The statistical analysis To check the statistical relationship between the plaintext and ciphertext produced using the proposed image cipher, we have done extensive statistical analysis by computing the image histograms, information entropy, correlation between the pair of plain and cipher images, and correlation between the adjacent pixels for a large number of images having widely different contents. In Table 1, we have shown the encryption of the two sample images i.e. image ‘Lena’ and an ‘all-zero’ image of size 200 X 200. Here the resultant images are shown after substitution using the quasigroup and permutation using the chaotic map for the two rounds. The secret key used for the encryption has also been men-


401

tioned in the first row of the table. In Figs 3 and 4 respectively, we have depicted the histograms of the plain and cipher images for the image ‘Lena’ and an ‘all-zero’ image for the first two rounds of encryption. Particularly in the first column-the histograms for the red, green and blue layers of the plain image have been shown however in the second and third columns the corresponding histograms of the cipher images after first and second rounds respectively have been depicted. It is very clear from the visual inspection of the histograms that the distributions of the pixel values in the cipher images are uniform/flat, random and independent of the original images. To quantify the uniformity of the pixel distribution in the cipher images, we have also computed the information entropy / Shannon entropy [37]. The information entropy quantifies the amount of information contained in data, usually in bits or bits/symbol. It is also the minimum message symbol length necessary to communicate the information. In other words, information entropy is a measure of disorder. For example, a long sequence of repeating characters has entropy 0, since every character is predictable and a truly random sequence has maximum entropy, since there is no way to predict the next character in the sequence. The information entropy H(m) of a message m is calculated by using the following relation H (m) =

N −1 2X

i=0

P(mi ) log2

1 (bits), P(mi )

(2)

where P(mi ) is the probability of occurrence of symbol mi in the message m and N is the number of bits required to represent a symbol in the message m. For a 24-bit colour image (I), the information entropy for each colour layer (Red, Green and Blue) is calculated using following expression H

R/G/B

(I) =

8 −1 2X

i=0

PR/G/B (Ii ) log2

1 PR/G/B (Ii )

(bits)

(3)

A truly random image (RI) has uniform distribution of pixel intensities in the interval [0, 255] i.e. PR/G/B (RI i ) = 1/256 for all i ∈ [0, 255], hence H R/G/B (RI) = 8 bits. In Table 2, we have given our computed results of information entropy for the image ‘Lena’ and an ‘all-zero’ plain images and their corresponding cipher images produced using the secret keys given in Table 1. It is clear that the information entropy is larger than 7.99 for all the cipher images produced through the proposed image cipher. We have also calculated the correlation between various pairs of plain and cipher images by computing the 2-dimensional correlation coefficients between various colour channels of the plain and cipher images. The 2D-correlation coefficients have been calculated using the following relation: 1 P H PW ¯ ¯ H×W i=1 j=1 (Ai, j − A)(Bi, j − B) (4) C AB = q 1 P H PW 1 P H PW 2 2 ¯ ¯ ( H×W i=1 j=1 (Ai, j − A) )( H×W i=1 j=1 (Bi, j − B) ) 1 PH PW 1 P H PW ¯ where A¯ = H×W i=1 j=1 Ai, j and B = H×W i=1 j=1 Bi, j . Here A represents one of the red (R), green (G) & blue (B) channels of the plain image, B represents one of the red (R), green (G) & blue (B) channels of the cipher image, A¯ & B¯ respectively, are the mean values of the elements of 2D matrices A & B and H & W respectively, are the height & width of the plain/cipher image. The results of our calculation of correlation coefficients for the image ‘Lena’ and an ‘all-zero’ images and their corresponding cipher images (shown in Table 1) have been given in Table 3. It is clear that the correlation coefficients between various channels of the plain image and cipher image are very small (or practically zero), hence the cipher images bear no clue about their corresponding plain images. To analyze the correlations of adjacent pixels (horizontally as well as vertically) in

402


Table 1 The encryption of images ‘Lena’ and ‘All-zero’.

The secret Key

x0 = 2.86295319532475 y0 = 4.56538639123458 K = 108.43745557666125 NS = 108; NR = 2 seed1 = 139; seed2 = 47 Quasigroup Image 256X256

Plain Image

After Substitution

NR = 1

After Permutation

Encrypted Image


403

Table 1 Continued.

After Substitution

NR = 2

After Permutation

Encrypted Image

Table 2 Information Entropy (bits). Red

Green

Blue

Plain Image Lena

7.2461

7.5410

6.9737

Cipher image Lena (NR=1)

7.9958

7.9948

7.9955

Cipher image Lena (NR=2)

7.9956

7.9958

7.9957

Plain image All Zero

0.0

0.0

0.0

Cipher image All zero (NR=1)

7.9947

7.9956

7.9953

Cipher image All zero (NR=2)

7.9956

7.9953

7.9954

the plain and cipher images, we have also computed the correlation coefficients for all the pairs of horizontally and vertically adjacent pixels in all plain and cipher images. This computation has been done by using the following formula: C= q

1 N

( N1

PN

PN

¯) i=1 (xi − x¯)(yi − y

¯ )2 )( N1 i=1 (xi − x

PN

i=1 (yi

(5) − y¯ )2 )

PN PN xi and y¯ = N1 i=1 yi . with x¯ = N1 i=1 Here xi and yi form ith pair of horizontally/vertically adjacent pixels and N is the total number of pairs of horizontally/vertically adjacent pixels. For an image of size W × H pixels N = (W − 1) H

404


400 200 0 0

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200 0 100

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0 0

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200

Fig. 3 The histograms of the plain image ‘Lena’ and its cipher images: The first column (from top to bottom) depicts the histograms of Red, Green and Blue channels of the plain image ‘Lena’. The second and third columns (from top to bottom) depict the histograms of Red, Green and Blue channels of cipher images of plain image ‘Lena’ after first and second rounds respectively. The secret key used for producing the cipher images is same as given in Table 1. Table 3 Correlation between plain and cipher images. Red

Green

Blue

NR = 1

Lena

Red

0.0047

0.0027

0.0044

Green

-0.0077

0.0028

-0.0033

Blue

0.0089

0.0036

-0.0023

0.0059

0.0018

NR=2 Red

-0.0024

Green

-0.0052

0.0039

0.0010

Blue

-0.0074

0.0028

0.0000

Red

0.0048

0.0069

0.0036

NR = 1

All Zero

Green

0.0048

0.0069

0.0036

Blue

0.0048

0.0069

0.0036

NR=2 Red

0.0051

0.0031

0.0037

Green

0.0051

0.0031

0.0037

Blue

0.0051

0.0031

0.0037


405

4

x 10 4

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0 0 100 4 x 10

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0 0

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200

Fig. 4 The histograms of an ‘all-zero’ plain image and its cipher images: The first column (from top to bottom) depicts the histograms of Red, Green and Blue channels of an ‘all-zero’ plain image. The second and third columns (from top to bottom) depict the histograms of Red, Green and Blue channels of cipher images of an ‘all-zero’ plain image after first and second rounds respectively. The secret key used for producing the cipher images is same as given in Table 1. Table 4 Correlation between adjacent pixels.

Plain Image Lena Cipher image Lena (NR = 1) Cipher image Lena (NR = 2) Plain image All Zero Cipher image All zero (NR = 1) Cipher image All zero (NR = 2)

Red

Green

Blue

Horizontal

0.9395

0.9385

0.8975

Vertical

0.9709

0.9706

0.9459

Horizontal

0.0022

0.0000

-0.0029

Vertical

-0.0055

0.0033

0.0027

Horizontal

0.0033

-0.0021

-0.0051

Vertical

-0.0057

0.0001

-0.0022

Horizontal

1.0000

1.0000

1.0000

Vertical

1.0000

1.0000

1.0000

Horizontal

-0.0019

0.0026

-0.0058

Vertical

0.0024

0.0022

0.0036

Horizontal

-0.0011

0.0023

-0.0014

Vertical

0.0015

0.0055

-0.0030

(for horizontally adjacent pixels) and N = (H − 1) W (for vertically adjacent pixels). The results of our calculation for the correlation coefficients between the horizontally and vertically adjacent pixels for the images ‘Lena’ and ‘All-zero’ and their corresponding cipher images are given in Table 4. We observe that adjacent pixels in plain images are highly correlated however for the pixels in cipher images

406


the correlation is almost zero hence the proposed image encryption technique removes the correlation between the adjacent pixels. 5 The key sensitivity analysis The extreme key sensitivity guarantees the security of a cryptosystem against the brute force attacks. The key sensitivity means the cipher image produced by the cryptosystem should be very sensitive to the secret key i.e., if we use two slightly different keys to encrypt the same image then two cipher images produced should be completely independent of each other. To test the key sensitivity of the pro-posed image cipher, we have first encrypted the plain image ‘Lena’ with a specific choice of the secret key and store the encrypted image as ‘Encrypted Image’. Now we make a small change in the one of the several parts of the secret key and produce the cipher image ‘Encrypted Image X’ and then we compare both the cipher images. The Table 5 gives the details of the various possibilities of minimal changes in the secret key and corresponding cipher images. To quantifying mutual independence of these encrypted images we have calculated the 2D-correlation and mutual information for the pair of encrypted images. The 2D correlation has been calculated in the same manner as done in the statistical analysis section above. However the mutual information between the two encrypted images I1 and I2 is calculated using MI(I1 ; I2 ) = H(I1 ) + H(I2 ) − H(I1 , I2 ), here H(I1 ), H(I2 ) are the information entropies calculated for the image I1 and I2 using the probability distributions P(I1 ), P(I2 ) respectively. However H(I1 , I2 ) is the joint information entropy of I1 and I2 calculated using the joint probability distribution P(I1 , I2 ). In Table 6, we have given the computed results of the 2D correlation coefficients and mutual information for the pair of encrypted images produced through very similar keys (as described in Table 5). It is very clear that the correlation between the cipher images produced using the very similar keys is negligible and also the mutual information between them is very low indicating that knowing the content of one cipher image I1 does not reveal any information about the other cipher image I2 and vice versa. Hence the proposed image cipher exhibits very high key sensitivity. 6 The differential analysis The differential analysis is the study of how small changes in the plaintext affect the resultant cipher-text with the application of the same secret key. It is generally done by implementing the chosen plaintext attack but now there are extensions which use known plaintext as well as ciphertext attacks also. For image cryptosystems a very common strategy to implement the differential analysis, is to make a slight change, usually one pixel, in the plain image and compare the two cipher images (obtained by applying the same key on two plain images having one pixel difference only). If a meaningful relationship can be established between the plain image and cipher image then it may further facilitates us in determining the secret key. However if the image cryptosystem produces significant, random and unpredictable changes in the ciphertext under one pixel change in the plain image then such differential analysis become inefficient. The two most common measures: NPCR (net pixel change rate) and UACI (unified average changing intensity) are used to test the vulnerability of the image cryptosystems against the differential analysis. The NPCR is used to measure the percentage number of pixels in difference of a particular colour channel in two cipher images obtained by applying the same secret key on two plain images having one pixel difference only. If Ci,R/G/B and C¯ i,R/G/B (where 1 ≤ i ≤ H and 1 ≤ j ≤ W, H is height and W is width j j and R, G and B represent red, green and blue channels) represent two cipher images then NPCR for

Vinod Patidar, N. K. Pareek, G. Purohit / Journal of Applied Nonlinear Dynamics 7(4) (2018) 393–412 50

100

150

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0

50

150

33.8

33.8

99.8

99.8

33.7

33.7

99.7

99.7

33.6

33.6

99.6

99.6

33.5

33.5

99.5

99.5

33.4

33.4

99.4

99.4

33.3

33.3

99.3

99.3

33.2

33.2

99.2

99.2

33.1

50

100

150

200

33.1 0

50

150

200

150

33.9

200 33.9

33.8

33.8

33.7

33.7

33.6

33.6

33.5

33.5

33.4

33.4

33.3

33.3

Test Number 0

50

100

Test Number

100

150

0

200

99.9

50

100

99.8

99.8

99.7

99.7 99.6

99.6 99.5

99.5

99.4

99.4

UACI (%)

99.9

99.3

99.3 33.2

99.2

33.2

99.2 33.1

0

50

0

50

100

150

200

150

200

0

50

0

50

Test Number

100

150

33.1 200

Test Number

100

99.9

33.9

200 33.9

33.8

100

150

33.8

33.7

33.7

33.6

33.6

33.5

33.5

33.4

33.4

33.3

33.3

99.9

99.8

99.8

99.7

99.7

99.6

99.6

99.5

99.5

99.4

99.4

99.3

99.3

UACI (%)

NPCR (%)

200

99.9

0

NPCR (%)

100

99.9

UACI (%)

NPCR (%)

0

407

33.2 99.2 0

50

100

99.2 200

150

33.2 0

50

Test Number

100

150

200

Test Number

Fig. 5 NPCR and UACI values for the plain image ‘Lena’: The first column (from top to bottom) depicts the distributions of NPCR values for Red, Green and Blue channels of plain Image ‘Lena’ while the second column (from top to bottom) depicts the distributions of UACI values for Red, Green and Blue channels of plain image ‘Lena’.

each colour channel is defined as: R/G/B

NPCR

=

R/G/B j=1 Di, j

PH PW i=1

W×H

× 100%,

(6)

  0 if Ci,R/G/B = C¯ i,R/G/B   j j  R/G/B . here Di, j =    1, if C R/G/B , C¯ R/G/B i, j i, j The NPCR value for two random images, which is an expected estimate for an ideal image cryp−LR/G/B ) × 100%, here LR/G/B is the number of bits used to tosystem, is given by NPCRR/G/B Expected = (1 − 2 represent the red, green or blue channels of the image. For a 24-bit true colour image (8 bit for each colour channel) LR/G/B = 8 hence NPCRR/G/B Expected = 99.6094%. The UACI, the average intensity difference of a particular channel between two cipher images Ci,R/G/B j and C¯ R/G/B , is defined as: i, j

UACI

R/G/B

R/G/B H W − C¯ i,R/G/B 1 X X Ci, j j = × 100%. R/G/B L W × H i=1 j=1 2 −1

(7)

The UACI value for two random images, which is an expected estimate for an ideal image cryptosystem is given by P2LR/G/B i(i + 1) 1 R/G/B × 100% (8) UACI Expected = . i=1 R/G/B R/G/B L 2 −1 22L Therefore for a 24-bit true colour image UACI R/G/B = 33.4635%. Expected

408


Table 5 Details of encrypted images produced for the key sensitivity analysis along with corresponding secret keys. Secret Key

1.

x0 = 2.86295319532475 y0 = 4.56538639123458 K = 108.43745557666125 NS = 108; NR = 2 seed1 = 139; seed2 = 47

2.

Quasigroup Image 256X256 1st and 2nd rows of the Quasigroup image are interchanged and the rest of the secret key remains same as in S. No. 1

Cipher Image

Encrypted Image Identification

Encrypted Image 1

3.

255th and 256th rows of the Quasigroup image are interchanged and the rest of the secret key remains same as in S. No. 1

Encrypted Image 2

4.

1st and 2nd columns of the Quasigroup image are interchanged and the rest of the secret key remains same as in S. No. 1

Encrypted Image 3

5.

255th and 256th columns of the Quasigroup image are interchanged and the rest of the secret key remains same as in S. No. 1

Encrypted Image 4

6.

128th and 129th rows of the Quasigroup image are interchanged and the rest of the secret key remains same as in S. No. 1

Encrypted Image 5

7.

128th and 129th columns of the Quasigroup image are interchanged and the rest of the secret key remains same as in S. No. 1

Encrypted Image 6

8.

seed1 value is changed by one unit and the rest of the secret key remains same as in S. No. 1

Encrypted Image 7

9.

seed2 value is changed by one unit and the rest of the secret key remains same as in S. No. 1

Encrypted Image 8

10.

NR value is changed by one unit and the rest of the secret key remains same as in S. No. 1

Encrypted Image 9

11.

NS value is changed by one unit and the rest of the secret key remains same as in S. No. 1

Encrypted Image 10

12.

K value is changed by 10−14 and the rest of the secret key remains same as in S. No. 1

Encrypted Image 11

13.

y0 value is changed by 10−14 and the rest of the secret key remains same as in S. No. 1

Encrypted Image 12

14.

x0 value is changed by 10−14 and the rest of the secret key remains same as in S. No. 1

Encrypted Image 13

We have done an extensive analysis to compute the NPCR and UACI for the proposed image cipher using two plain images ‘Lena’ and ‘All-zero’ and the secret key given in the first row of the Table 1. Particularly we have randomly chosen 200 different pixels (one at a time, including the very first and very last pixels of the image) in each plain image and changed one of the R, G, B intensity values by one unit only and computed the NPCR and UACI as explained above for all 200 cases each for both the plain images. The computed results of NPCR and UACI have been depicted in Figs. 5 and 6 respectively for the ‘Lena’ and ‘all-zero’ images. It is clear that the NPCR and UACI values are distributed in a small interval around the ideal values 99.6094 and 33.4635 (shown by the horizontal lines) respectively. Hence the proposed image cipher shows extreme sensitivity on the plaintext and not vulnerable to the differential attacks like: chosen plaintext, known plaintext and adaptive chosen plaintext attacks.


409

Table 6 Correlation and mutual information between ‘Encrypted Image’ and ‘Encrypted Image X’ (correlate with Table 5). Correlation

Encrypted Image X

Mutual Information (bits)

R

G

B

Encrypted Image 1

-0.0050

0.0068

0.0021

0.4505

Encrypted Image 2

-0.0058

-0.0047

-0.0086

0.4491

Encrypted Image 3

-0.0015

-0.0037

0.0025

0.4480

Encrypted Image 4

0.0007

0.0061

0.0048

0.4484

Encrypted Image 5

0.0015

0.0037

0.0039

0.4470

Encrypted Image 6

0.0033

0.0009

-0.0031

0.4495

Encrypted Image 7

-0.0024

-0.0009

-0.0005

0.4499

Encrypted Image 8

-0.0056

0.0019

-0.0075

0.4482

Encrypted Image 9

0.0013

0.0016

0.0061

0.4517

Encrypted Image 10

-0.0023

-0.0016

0.0056

0.4515

Encrypted Image 11

0.0046

-0.0066

-0.0052

0.4471

Encrypted Image 12

-0.0016

-0.0024

-0.0010

0.4478

Encrypted Image 13

-0.0060

0.0089

-0.0082

0.4496

50

100

150

200

0

150

200 33.8

99.8

99.8

33.7

33.7

99.7

99.7

33.6

33.6

99.6

99.6

33.5

33.5

99.5

99.5

33.4

33.4

99.4

99.4

33.3

33.3

99.3

99.3

33.2

33.2

99.2

99.2

33.1

50

100

150

200

0

50

100

33.1 0

150

200

150

33.9

200 33.9

33.8

33.8

33.7

33.7

33.6

33.6

33.5

33.5

33.4

33.4

33.3

33.3

33.2

33.2

Test Number

50

100

Test Number 150

200

0

50

100

99.8

99.8

99.7

99.7

99.6

99.6

99.5

99.5

99.4

99.4

99.3

99.3

99.2

99.2 0

50

100

150

UACI (%)

99.9

99.9

NPCR (%)

100

33.8

0

33.1

200

0

Test Number 50

100

150

99.9

99.9

99.8

99.8

99.7

99.7

99.6

99.6

99.5

99.5

99.4

99.4

99.3

99.3

50

100

150

0

50

100

150

33.9

200 33.9

33.8

33.8

33.7

33.7

33.6

33.6

33.5

33.5

33.4

33.4

33.3

33.3 33.2

33.2 99.2 0

50

100

Test Number

150

99.2 200

33.1 200

Test Number

200

UACI (%)

0

NPCR (%)

50

99.9

UACI (%)

NPCR (%)

0 99.9

0

50

100

150

200

Test Number

Fig. 6 NPCR and UACI values for the “all-zero’ plain image : The first column (from top to bottom) depicts the distributions of NPCR values for Red, Green and Blue channels of ‘all-zero’ plain Image while the second column (from top to bottom) depicts the distributions of UACI values for Red, Green and Blue channels of ‘all-zero’ plain image.

410


From all the analysis results described in Sections 4 to 6, we may infer that there is no significant difference in the results obtained after the first and second rounds. It is due to the fact that the proposed technique has excellent capabilities of producing confusion and diffusion in only one round therefore not much qualitative/quantitative changes appear in the subsequent rounds. However from the security point of view, it is always advisable to have intertwined confusion and diffusion to make any kind of cryptanalysis infeasible. Therefore we recommend minimum two number of rounds to have foolproof security. It is true that increasing number of rounds will affect the encryption/decryption rate (encryption/decryption time). On an Intel Core 2 Duo 2.1 GHz CPU running on Windows 7 and using MATLAB 7.2 programming, the proposed technique could encrypt/decrypt at an average speed of 15.3 Mbps and 11.6 Mbps with only one and two rounds respectively. However, we strongly believe that further improvement in the rate of encryption/decryption of algorithms can be achieved by adopting proper optimization techniques (for which we did not pay much attention here). 7 The keyspace analysis The secret key in the proposed image encryption technique is divided into two parts. First part consists of a set of three floating point numbers and four integers (x0 , y0 , K, NS , NR, seed1, seed2), where (x0 , y0 ) ∈ (0, 2π), K can have any real value greater than 18.0, NS is an integer value preferably greater than 100, NR is an integer value between 1 and 16, which is treated as no. of encryption rounds, seed1 and seed2 are integers between 0 to 255 and used as the seed values for the initiation of the substitution. The second part is a 256 × 256 grey image whose pixel values form a 256 × 256 Latin square. The key space (i.e., the total number of different keys that can be used in the encryption/decryption) for the proposed image encryption scheme can be calculated as follows: (i) By following the discussion in Patidar et al. [14] the key space for the chaotic standard map parameters is (6.28)3 × 1042, (ii) total possible values of NS is 103 , (iii) total possible values of NR is 15, (iv) total possible values of seed1 and seed2 are 256 each and the most important the minimum possible combination of quasigroup QG image of 256 × 256 is 10101724 . Hence the total key space of the proposed image encryption algorithm is greater than 2.434 × 10101769 . 8 Conclusion In a large number of proposals on chaos based image encryption schemes, it has been observed that these schemes offer a good combination of performance and security features. In most of such schemes control parameters or initial conditions of the chaotic systems have been used as part of the secret key. However, chaotic systems offer desirable properties only for the certain ranges of such pa-rameters/initial conditions; the key space available for such image encryption techniques is greatly reduced and opens possibilities for a brute force attack. With an objective to increase the available key space and retaining all the desirable properties offered by chaos based image encryption schemes, we have proposed a simple, efficient and robust image encryption scheme based on the combined applications of quasigroups and chaotic dynamical systems. Our analysis results show that the proposed image cipher possesses a very large key space (which is practically impossible to break through brute force attack) as well as all other desirable properties which an ideal image encryption system should have. We believe that it is one of the very first steps taken and further research will open new avenues in the area of combined application of chaos and quasigroups (Latin squares) in image encryption.


411

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On Some Properties of Memristive Lorenz Equation – Theory and Experiment P. Saha1 , D. C. Saha2 , A. Ray3 , A. Roy Chowdhury4† 1

2 3 4

Department of Physics, B.P. Poddar Institute of Management & Technology, 137 VIP Road, Kolkata700052, India Department of Physics, Prabhu Jagatbandhu College, Andul, Howrah-711302, India Department of Physics, Gour Mahavidyalaya, Malda-732142, India High Energy Physics Division, Department of Physics, Jadavpur University, Kolkata-700032, India Submission Info Communicated by A.C.J. Luo Received 19 April 2017 Accepted 19 September 2017 Available online 1 January 2019 Keywords Memristor Lorenz system Bifurcation Line fixed point

Abstract A memristive version of Lorenz equation is proposed and then the equivalent analogue circuit is constructed. In the experimental realization we have used the Op-amp equivalent of a memristor. Starting from the basic stability analysis, the formation of periodic orbits to attractors and the generation of bifurcation scenario, all are shown to depend on the memristive parameters very significantly. As a whole, the memristor has a controlling effect on the system. The overall system being four dimensional, is hyperchaotic and shows some very interesting transitions. Our experimental data supports the numerical simulations. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In recent years, there have been various activities to understand the effect of memristor on nonlinear dynamical systems. It was suggested by L. O. Chua in 1971 [1] that there should be a fourth fundamental electronic element other than the three well known components - resistor, capacitor and inductor. These electronic elements give us the relationships between pairs of four quantities namely – charge (q), current (i), voltage (V ) and flux (φ ). A missing link between charge and flux was introduced by Chua as the “Memristor”. Recently in 2008, Hewlett-Packard announced a possible fabrication of memristors in Nature [2], where, a new nanometer-scale electric switch was realized which remembers whether it is “on” or “off” after its power is switched off. The fabrication of memristors attracted the attention of researchers due to its wide applicability in various systems. Phenomena in nanoscale systems such as thermistor [3] and spintronic devices [4] can be explained by memristors. Memristors can also simulate the behaviour of biological synapses, adaptive behaviour of unicellular organisms [5] and learning and associative memory [6]. Although † Corresponding



414

P. Saha et al. / Journal of Applied Nonlinear Dynamics 7(4) (2018) 413–423

the component is not yet commercially produced yet it has been observed that a very interesting Opamp combination can replicate the working of a memristor. Recently, Chua’s diode was replaced by monotone increasing piecewise linear memristor and various memristors based on chaotic circuits was also proposed [7, 8]. So it is evident that the unique memory of initial state of the memristors creates complex dynamical behaviour which is different from the well known chaotic systems. Memristors are also applicable in the field of chaos control [9], synchronization [10] and image enecryption [11]. Global anti-synchronization of two chaotic delayed memristive neural networks was also shown [12]. As such there have been several attempts to get the effect of memristor in some important nonlinear electronic circuits. On the other hand, there exists a good many nonlinear dynamical systems whose chaotic and periodic behaviours have been studied in detail. Interestingly almost all of these systems can be represented with the help of electronic circuits. Here in this communication, we have proposed a new nonlinear equation by incorporating memristive effect in the age old Lorenz system [13]. The resulting nonlinear dynamical system is governed by four nonlinear equations and as such show hyperchaotic behaviour [14]. Subsequently we have constructed the electronic circuit corresponding to our system. The ensuing data is then collected via a National instrument NI-DAC6000. We have analyzed the role of memristive parameters in the generation of periodic orbits, bifurcation patterns and the overall stability of the dynamics. Our findings strongly suggest that the memristor plays the role of a controller. All these numerical simulations are corroborated by experimental data. 2 Formulation The proposed memristive Lorenz equation can be written as x˙ = σ (y − x) y˙ = (α + β u2 )(r − z)x − y z˙ = xy − bz

(1)

u˙ = (r − z)x where (α + β u2 ) represents the effect of the flux controlled memristor. As it is evident that the unstable fixed point is {x = 0, y = 0, z = 0 and u = u}. The corresponding Jacobian and the relevant eigenvalues are; −σ (α + β u2 )r J = 0 r

σ −1 0 0

0 0 −b 0

0 0 0 0

(2)

along with;

λ1 = 0 λ2 = −b 1+σ 1√ λ3 = − P + 2 2 1+σ 1√ − λ4 = − P 2 2

(3)


415

Fig. 1 Variation of λ3 with u and β .

where

P = 1 − 2σ + σ 2 + 4α rσ + 4σ rβ u2 ,

The character of these eigenvalues changes with that of P. To ascertain this we have to set r = 28.0, σ = 10.0, b = 8/3, α = 1.0, but β is varied from −1 to +1. Also since u is arbitrary, this was varied between (−4, 4) so that λ3 can be visualized as in Fig-1. It is evident from this figure that there are some regions where this becomes complex – that is a Hopf bifurcation occurs. Similarly the variation of λ4 has been displayed in Fig-2. To proceed further we numerically integrate equation (1) whence we obtain different form of the corresponding attractors. For r = 28.0, σ = 10.0, b = 8/3, α = 1.0, β = 0.05 we get the usual form given in Fig-3(a-b). On the other hand for r = 60.0, σ = 10.0, b = 8/3, α = 1.0, we get the hyperchaotic attractors whose 2D projections are shown in Fig-4. 3-D projections of the attractor showing the hyperchaotic form in the (x, y, z) and (u, x, y) space is given in Fig-5. These figures can be later compared with the experimental results as described in Section-3. To understand the role of the memristor further we have plotted the bifurcation diagram with respect to the parameters. First we consider the ordinary Lorenz case where σ = 10.0, b = 8/3, α = 1.0, β = 0.001 and ‘r’ varies. This is shown in Fig-6, where we can clearly see the existence of both periodic and chaotic states. Next the variation by changing the value of β is observed. In Fig-7, where β = 0.05 one can observe that a greater part of periodic window disappears. After r = 109 the attractor disappears due to boundary crisis. Same thing happens after r = 56 when greater value of β = 0.5 is chosen and this is quite vivid in Fig-8. The situation in Fig-7 is further clarified in Fig-9, where for r = 61.5, σ = 10.0, b = 8/3, α = 1.0, β = 0.05 we show period two orbit. It can be mentioned here that for r = 61.95 in the original Lorenz we usually get chaotic state. This is actually a controlling mechanism which is generated due to the presence of memristor. To clarify the situation further we have computed the Lyapunov exponents for various parameter combination and have obtained bi-parametric plots. These are exhibited in figures-10 to 13.

416


Fig. 2 Variation of λ4 with u and β .

60

45 40

40

35 20 y

y

30 0

25 20

−20

15 −40 10 −20

−10

0 x

10

20

−50

0 x

50

Fig. 3 2D phase diagrams of the system at r = 28.0, σ = 10.0, b = 8/3, α = 1.0, β = 0.001.

In Fig. 10, we fixed σ = 10.0, b = 8/3, α = 1.0, but β and r are changing. In the corresponding diagram the darker zone indicates chaotic region whereas lighter zone stands for periodic states. Similarly for the case where σ is not constant, is shown in figure-11. These two diagrams shows the variation of first Lyapunov exponent, whereas for the system to be hyperchaotic one should examine also the second Lyapunov exponent. Variation of this is shown in figures 12 and 13 and it is observed that even the second Lyapunov becomes positive in various situations signalling a hyperchaotic state.


417

(b)

(a)

100

150

90 100

80

70

y

z

50

0

60

50

40 Ŧ

Ŧ50

30

Ŧ

Ŧ100 Ŧ Ŧ40Ŧ

Ŧ30Ŧ

Ŧ20Ŧ

Ŧ10

0 x

10

20

30

20 Ŧ100 Ŧ

40

Ŧ50Ŧ

0

50

100

150

x

(a)

(b)

Fig. 4 2D representation of the attractor in (x, y) & (x, z) plane at α = 1.0, β = 0.05, σ = 10, r = 60 and b = 83 . 100

25

90

20

80

15 10

70

u

z

5

60

0

50

Ŧ5

Ŧ

40

Ŧ10 Ŧ

30

Ŧ15 Ŧ

20 150

Ŧ20 Ŧ 150

100

40 50

20 0 Ŧ

y

100

40 50

20 0

0 Ŧ50 Ŧ

Ŧ

Ŧ100 Ŧ

Ŧ40

(a)

0 Ŧ50 Ŧ

Ŧ20 x

y

Ŧ20 Ŧ

Ŧ100 Ŧ Ŧ40 Ŧ

x

(b)

Fig. 5 3D representation of the attractor in (x,y,z) & (x,y,u) plane with parameter values α = 1.0, β = 0.05, σ = 10.0, r = 60.0 and b = 8/3.

3 Electronic Circuit The electronic circuit constructed is displayed in Fig. 14, where we have used four Op-amps TL084CN along with two AD633JN multipliers . In the present circuit, the novel component is the memristor U4 which is connected between the multiplier at U3 and the op-amp at U1B . The circuit also consists of a good number of capacitors, resistances and voltage sources. The memristor U4 is actually the circuit shown in Fig. 15. The output is observed with the help of an oscilloscope. Also the data was collected with the help of Analogue to digital converter NI-6000, DAC. The data so collected can be analyzed for various purposes. The form of the attractors and periodic orbits, as seen on the oscilloscope screen, are given in Fig. 16 & 17 along with their time series. A scaled expression of equation (1) is used for electronic circuit representation. The scaling used is given by y z x and Z = √ X=√ ,Y=√ as r as r as r

(4)


Fig. 6 Bifurcation diagram w.r.t ‘r’ for β = 0.001 where σ = 10.0, α = 1.0 , b = 38 .

Fig. 7 Bifurcation diagram w.r.t ‘r’ for β = 0.05 where σ = 10.0, α = 1.0 , b = 83 . 35 30 25 20 x

418

15 10 5 0 10

15

20

25

30

35 R

40

45

50

55

60

Fig. 8 Bifurcation diagram w.r.t ‘r’ for β = 0.5 where σ = 10.0, α = 1.0 , b = 38 .


419

100 80 60 40

y

20 0 Ŧ20 Ŧ40 Ŧ60 Ŧ80 Ŧ30

Ŧ20

Ŧ10

0 x

10

20

30

Fig. 9 Period 2 orbit for r = 61.95 with σ = 10.0, b = 83 , α = 1.0, β = 0.05. 0.05 1 0.1 0.8

β

0.15 0.2

0.6

0.25

0.4

0.3 0.2 0.35 0 0.4 Ŧ0.2

0.45 0.5 20

30

40

50

60

70

80

90

100

r

Fig. 10 Parametric space plot w.r.t first Lyapunov exponent for r and β where α = 1.0, σ = 10.0 and b = 83 .

Here ‘as ’ is the scale parameter is to be chosen. With the scaling-(4), the system dynamics are governed by dX = σ (Y − X ), dt dY 1 = as rW (φ )( − Z)X −Y, dt as dZ = XY − bZ. dt

(5)

Based on the magnitude of oscillation, we chose as = 13 . This makes the oscillator to oscillate within acceptable voltage range. The oscillator parameters σ , r and β for the memristive Lorenz circuit are controlled with circuit resistors R2 ,R8 , R10 , R11 and R12 . Among these five resistors two are shown in Fig. 14 and rest are shown in Fig. 15. The relation between the resistances and the parameters are given below,

σ=

100kΩ 10kΩ R12 + R13 100kΩ , b= , as rα = and as rβ = 10kΩ( ). R8 R2 R11 100R12 R11

420

P. Saha et al. / Journal of Applied Nonlinear Dynamics 7(4) (2018) 413–423 1.4 15

1.2

20

1

25 r

0.8 30 0.6 35 0.4 40 0.2 45 0 50 2

3

4

5

6

7

8

9

10

σ

Fig. 11 Parametric space plot w.r.t first Lyapunov exponent for σ and r where α = 1.0, β = 0.05 and b = 38 . 0.05 0.4 0.1 0.15

0.2

0.2 β

0 0.25 0.3

Ŧ0.2

0.35 Ŧ0.4

0.4 0.45

Ŧ0.6

0.5 20

30

40

50

60

70

80

90

100

r

Fig. 12 Parametric space plot w.r.t second Lyapunov exponent for r and β where α = 1.0, σ = 10.0 and b = 38 . 0.2 15 0.1 20 0 25 R

−0.1 30 −0.2 35 −0.3 40 −0.4

45 50

−0.5 2

3

4

5

σ

6

7

8

9

10

Fig. 13 Parametric space plot w.r.t second Lyapunov exponent for σ and r where α = 1.0, β = 0.05 and b = 83 .


421

The memristive element is described through the functional relation I = W (φ )V where, W (φ ) is given as, W (φ ) = α + β φ 2 . This is described in Fig. 15. The transfer function of electronic circuit is given by ´ ( R101 C V d τ )2 R12 R13 1 W (φ ) = −{ + }V. R11 100R12 R11

(6)

We have taken C1 = C2 = C3 = C4 = 10nF. For fixing the parameter value at α = 1.0, β = 0.01, σ = 10.0, r = 60.0 and b = 38 , we show the resistance values in Figs. (14) and (15). Later on we have changed these resistance according to change the parameter values. C1 C2

0.01µF R2

0.01µF

39kȍ

R4

V2

R6

100kȍ U2 X1 X2 Y1 Y2

1112

R1

VS+ W Z VS-

VU1A

U3 X1 X2 Y1 Y2

2

10kȍ

AD633JN

1 3

VS+ W Z VS-

U4 1

11 2

U1B

0.01µF U1C

11

R5

6

Memristor

AD633JN

TL084CN

4

C3 100kȍ

7

100kȍ

5

8

TL084CN

4

10kȍ TL084CN

4

12 V

U1D

13

10

V1 V3 3V

11

R8

9

14 12

R7

R9

100kȍ

4

TL084CN

100kȍ

Fig. 14 Circuit diagram of the system given by Equation-(1).

8

U7C

9

10

4

11

R6 2kȍ

R7 2kȍ

TL084CN R5 500ȍ

U6 X1 X2 Y1 Y2

1

C3

4

VS+ W Z VS-

R4 30.0kȍ

AD633JN

100nF IC=0V 5%

U7A

3

11

R2

U7B U5

1 6 7

X1 X2 Y1 Y2

TL084CN

AD633JN

100kȍ

2 11

TL084CN

5 4

VS+ W Z VS-

R3 29.4kȍ

V2 12 V

V1 -12 V

2

Fig. 15 Details of the construction of memristor U4 .

422


(a)

(b)

(c)

(d)

Fig. 16 Phase plot of the attractors on the oscillator screen for α = 1.0, β = 0.01, σ = 10.0, r = 60.0 and b = 38 .

(a)

(b)

(c)

Fig. 17 Phase plot of the periodic attractors on the oscillator screen for α = 1.0, β = 0.05, σ = 10.0, r = 61.95 and b = 38 .

4 Conclusion In our above analysis we have analyzed a new proposed memristive Lorenz equation. The system is seen to be hyperchaotic and shows strong variations with respect to the memristive parameters. The memristor introduced works as a controlling device which is evident from the periodic state of the system at the chaotic parametric values of the original Lorenz system. Such situations are also depicted in bifurcation diagrams and Lyapunov patterns. Chaotic signals derived from such memristive circuits will enable to construct complex and unpredictable time domain signals which can help in secure communication and data encryption. In this connection it may be added that we can construct another form of this memristive Lorenz


423

if the following equation is considered instead of (1) x˙ = σ (y − x), y˙ = rx − y −W (u)xz,

(7)

z˙ = xy − bz, u˙ = xz. Even then many interesting events can be visualized. But the important point is that no effect of the memristor can be seen in the linear stability analysis (that is the effect of α and β ). But numerically we can still obtain the form of the attractors, periodic orbits, bifurcation etc. although the reason behind these cannot be ascertained very clearly and needs further analysis. Acknowledgement PS is thankful to SERB (DST, Govt. of India) for a research project and ARC is thankful to UGC (Govt. of India) for a UGC-BSR faculty fellowship which made this work possible. References [1] Chua, L.O. (1971), Memristor - The missing circuit element, IEEE. Trans. on Circuit Theory, 18, 507–519. [2] Strukov, D. Snider, G. Stewart, G. and Williams, R. (2008), The Missing Memristor Found, Nature, 453, 80–83. [3] Sapoff, M. and Oppenheim, R.M. (1963), Theory and Application of self heated Thermistors, Proc. IEEE, 51, 1292–1305. [4] Pershin, Y.V.and Ventra, M. Di (2008), Spin Memristive Systems: Spin Memory effects in Semiconductor Spintronics, Phys. Rev. B, Condens. Matter, 78, 113309/1–4. [5] Pershin, Y.V. Fontaine, S. La and Ventra, M. Di (2010), Memristive model of Amoeba Learning, Phys. Rev E, 80, 1335–1350. [6] Pershin, Y.V. and Ventra, M. Di (2010), Experimental demonstration of Associative Memory with Memristive Neural Networks, Neural Networks, 23, 881–886. [7] Itoh, M. and Chua, L. O. (2008), Memristor Oscillators, International Journal of Bifurcation and Chaos, 18(11), 3183–3206. [8] Chua, L.O., Kang, Mo Sung (1976), Memrisive devices and systems, Proceedings of the IEEE, 64(2), 209–223. [9] Bao, B. Liu, Z. and Jian-Ping, X. (2010), Transient chaos in smooth memristor oscillator, Chinese Phys. B, 19, 030510. [10] Wen, S. Zeng, Z.and Huang, T.(2012), Adaptive synchronization of memristor-based Chua’s circuits, Phys. Lett. A, 376, 2775–2780. [11] Lin, Z.H. and Wang, H.X.(2010), Efficient Image Encryption Using a Chaos-based PWL Memristor, IETE Tech. Rev., 27, 318–325. [12] Zhang, G. Shen, Yi and Wang, L. (2013), Global anti-synchronization of a class of chaotic memristive neural networks with time-varying delays, Neural networks, 46, 1–8. [13] Lorenz, E. N. (1963), Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130-141. [14] Cafagna D., Grassi G.(2003), New 3D-scroll attractors in hyperchaotic Chuas circuits forming a ring, International Journal of Bifurcation and Chaos, 13(10), 2889–2903.



Nonlinear Integral Inequalities WITH Parameter and Applications Taoufik Ghrissi†, M. A. Hammami Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax Route Soukra, BP 1171, 3000 Sfax, Tunisia Submission Info Communicated by A.C.J. Luo Received 12 May 2017 Accepted 4 October 2017 Available online 1 January 2019 Keywords Nonlinear inequalities Nonlinear systems Lyapunov function Bounded solutions

Abstract We discuss the problem of establishing dissipative estimates for certain differential equations for which the usual methods do not work. The aim of this paper are some new nonlinear integral inequalities leading to suitable uniform (with respect to time and the parameter ε ) bounds on the solutions to problems x(t) ˙ = f (t, x(t)) + gε (t, x(t)), t ≥ 0 where f , gε : R+ × Rn → Rn are supposed to be piecewise continuous in time, locally Lipschitz in x, for any fixed ε ≥ 0. These problems are seen as perturbation to x(t) ˙ = f (t, x(t)), t ≥ 0. Furthermore, some examples are given to illustrate the applicability of the obtained results. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The aim of this paper is to present a new Gronwall-type inequality, somehow already implicitly exploited in some recent papers (see [1, 2]), which allows to establish dissipativity when the standard techniques do not apply. The main feature of this method, compared to the usual ones, is that it is not focused on a single differential inequality like ψ ′ (t) + νψ (t) ≤ C, but rather on the whole family of such inequalities, depending on a parameter ε > 0. In fact, we use different values of the parameter ε in different regions of the phase-space and, even in order to verify estimate for a single trajectory, we need to consider many differential inequalities corresponding to different values of the parameter ε . To be more precise, we study a family of differential inequalities of the form ψ ′ (t) + εψ (t) ≤ Cε α [ψ (t)]β +C, (1) depending on a small parameter ε > 0. Here C > 0 and α > β ≥ 1. Obviously, the solutions of the ordinary differential equation associated with (1), for any fixed ε > 0, may blow up in finite time (if † Corresponding



426

Taoufik Ghrissi, M. A. Hammami / Journal of Applied Nonlinear Dynamics 7(4) (2018) 425–436

β > 1) and, for this reason, any single inequality is not strong enough to give the dissipative estimate for ψ . However, as we will see, if the function ψ satisfies simultaneously (1) for all ε ∈ (0, ε0 ], for some ε0 > 0, then the conclusion ψ (t) ≤ Q(ψ (0))e−ν t +C⋆ , is still true, for some ν > 0, C⋆ ≥ 0 and some nonnegative increasing function Q (which can be explicitly written in terms of the parameters α , β and C), so yielding the desired dissipativity. The problem of asymptotic stability analysis of nonlinear time-varying systems has attracted the attention of many researchers and has produced a vast of important results ([3–10] and the references therein). This fact motivated to study systems whose desired behavior is asymptotic stability about the origin of the state space or a close approximation to this, e.g., all state trajectories are bounded and approach a sufficiently small neighborhood of the origin [7] and references therein. Quite often, one also desires that the state approaches the origin (or some sufficiently small neighborhood of it) in a sufficiently fast manner. In 1919 Gronwall [11] introduced the famous Gronwall inequality in the study of the solution of differential equations. Since then, a lot of contributions have been achieved by many researchers (see [12–14]). These inequalities can be used in the study of existence, uniqueness, and stability properties to the solutions of differential equations. In this paper, We try to prove some new Gronwall-type lemmas to find uniform bounds to solutions of perturbed systems of the form: x(t) ˙ = f (t, x(t)) + gε (t, x(t)),t ≥ 0 where f , gε : R+ × Rn → Rn are supposed to be piecewise continuous in time, locally Lipschitz in x, for any fixed ε ≥ 0. As application, we will consider many cases and give some examples to show the utility of our results. 2 General definitions We start by introducing some notations and definitions that will be employed throughout the paper. Consider the time varying system described by the following equation: x˙ = f (t, x),

(2)

where f : R+ × Rn −→ Rn is a piecewise continuous in t and locally Lipschitz in x on R+ × Rn . For all x0 ∈ Rn and t0 ∈ R, we will denote by x(t;t0 , x0 ), or simply by x(t), the unique solution of the system (2) at time t0 starting from the point x0 . Unless otherwise stated, we assume throughout the paper that the functions encountered are sufficiently smooth. We often omit arguments of functions to simplify notation, k.k stands for the Euclidean norm vectors. We recall the following definitions (see [10]). Definition 1. The solutions of system (2) are uniformly bounded if there exists R > 0, such that for all R1 > 0, there exists a T = T (R1 ), such that for all t0 ≥ 0 kx0 k ≤ R1 ⇒ kx(t)k ≤ R,

∀t ≥ t0 + T.

Let x = 0 be an equilibrium point for the nonlinear system (2). The origin is an equilibrium point for (2), if f (t, 0) = 0,

∀t ≥ 0.


427

Definition 2. x = 0 is said to be globally exponentially stable if there exist K, λ > 0, such that all trajectories satisfy kx(t)k ≤ Kkx0 ke−λ (t−t0 ) , ∀ x0 ∈ Rn , ∀t ≥ t0 ≥ 0. Suppose that the jacobian matrix [∂ f /∂ x] is bounded on Rn , uniformly in t. Assume that the system (2) is globally exponentially stable, then by a classical Theorem in [10], there is a continuously differentiable Lyapunov function V : [0, +∞[×Rn −→ R+ , that satisfies the following conditions for some positive constants c1 , c2 , c3 , c4 and for all x ∈ Rn , t ≥ t0 : c1 kxk2 ≤ V (t, x) ≤ c2 kxk2 , V˙ (t, x) ≤ −c3 kxk2 , and k

∂V (t, x)k ≤ c4 kxk. ∂x

3 Some Gronwall-type lemmas with parameter There exist many Lemmas which carry the name of Gronwall’s Lemma. A main class may be identified is the integral inequality. The original Lemma proved by Gronwall [11], was the following: Lemma 1. (Gronwall) Let z : [a, a + h] → R be a continuous function that satisfies the inequality ˆ x (A + Mz(s))ds 0 ≤ z(x) ≤

(3)

a

for all a ≤ x ≤ a + h, where A, M ≥ 0 are constants. Then 0 ≤ z(x) ≤ AheMh

(4)

for all a ≤ x ≤ a + h. The above Lemma can be formulated by the following famous inequality, which is called the Gronwall inequality: Let u(t) be a continuous function defined on the interval [t0 ,t1 ] and ˆ t u(t) ≤ a + b u(s)ds,

(5)

t0

where a and b are nonnegative constants. Then, for all t ∈ [t0 ,t1 ], we have u(t) ≤ aeb(t−t0 ) .

(6)

After more than 20 years, Bellman [13] extended the last inequality, which reads in the following: Let a be a positive constant, u(t) and b(t), t ∈ [t0 ,t1 ] be real-valued continuous functions, b(t) ≥ 0, satisfying

428


u(t) ≤ a +

t

ˆ

b(s)u(s)ds, t ∈ [t0 ,t1 ].

(7)

t0

Then, for all t ∈ [t0 ,t1 ], we have ´t

u(t) ≤ ae

t0

b(s)ds

.

(8)

The somewhat more general extensions of the original Gronwall inequality can be found in [12–14]. Next, we recall the following results: 1 Proposition 2. (see [15]) Let α > β ≥ 1 and γ ≥ 0 be such that αβ −1 −1 < 1+γ . Let ψ be a nonnegative absolutely continuous function on [0, ∞) which fulfills, for some K ≥ 0, Q ≥ 0, ε0 > 0 and every ε ∈ (0, ε0 ], the differential inequality ψ ′ (t) + εψ (t) ≤ K ε α [ψ (t)]β + ε −γ q(t), (9)

where q is any nonnegative function satisfying sup

t+1

ˆ

t≥0

q(y)dy ≤ Q.

(10)

t

Then, there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0 such that

ψ (t) ≤ R0 , ∀t ≥ tR ,

(11)

whenever ψ (0) ≤ R. Both R0 and tR can be explicitly computed. Definition 3. (see [15]) Let P = {ς , α , β , γ , K,C} an assigned structural set of nonnegative parameters, with ς ∈ (0, 1] and

α − 1 > (1 + γ )(β − 1) ≥ 0.

(12)

We denote by T the space of nonnegative locally summable functions on R+ such that khkT = sup t≥0

ˆ

t+1

h(s)ds < ∞.

(13)

t

If h ∈ T , the following inequality holds for every ε ∈ (0, 1] and t ≥ τ ≥ 0,

ε

t

ˆ τ

8 e−ε (t−s) h(s)ds ≤ khkT . 5

(14)

Proposition 3. (see [16]) Let k ∈ T with kkkT ≤ K and ϕ : R+ −→ R+ a continuous function satisfying the integral inequality −ε t

ϕ (t) ≤ Re

+ε

α

ˆ

t

e−ε (t−s) k(s)[ϕ (s)]β ds +Cε −1−γ

(15)

0

for every ε ∈ (0, ς ], every t ≥ 0 and some R ≥ 0. Then, there exist two strictly positive constants ω , Λ and an increasing function ℑ : R+ −→ R+ such that

ϕ (t) ≤ ℑ(R)e−ω t + Λ C.

(16)


429

Remark 1. The constants ω and Λ are independent of R, and can be explicitly calculated in terms of the structural set P (see [18] for the exact values). In particular, once the other parameters are fixed, Λ becomes independent of C sufficiently small. Consequently, we get exponential decay of rate ω when C = 0. Remark 2. (see [16]) Let h ∈ T , if

ψ ′ (t) + εψ (t) ≤ ε α k(t)[ψ (t)]β + h(t)ε −γ ,

(17)

then, an application of the Gronwall Lemma provides that −ε t

ψ (t) ≤ ψ (0)e

+ε

α

t

ˆ 0

8 e−ε (t−s) k(s)[ψ (s)]β ds + khkT ε −1−γ . 5

(18)

Note that, we can use the last proposition in this situation. 4 New generalizations Proposition 4. Let α > β ≥ 1 and γ ≥ 0 be such that satisfying

lim H(r) r→∞ r

β −1 α −1

0 and every ε ∈ (0, ε0 ], the differential inequality:

ψ ′ (t) + εψ (t) ≤ K ε α H(ψ (t)) + ε −γ q(t),

(19)

where q is any nonnegative function satisfying sup t≥0

ˆ

t+1

(20)

q(y)dy ≤ Q.

t

Then, there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0, such that

ψ (t) ≤ R0 , ∀t ≥ tR ,

(21)

whenever ψ (0) ≤ R. Both R0 and tR can be explicitly computed. Proof. The hypothesis on q implies that, for any t ≥ 0, ˆ t+τ q(y)dy ≤ Q(1 + τ ), ∀τ > 0.

(22)

t

Due to the assumptions on α , β , γ , we can select υ ∈ (0, 1) satisfying the inequality: 1 − υ > max{β − αυ , γυ }.

(23)

Calling ω = 1 − γυ > υ , we consider the function J(r) = −ω r1−γυ −υ + ω Kr−γυ −υ H(r).

(24)

υ

As lim J(r) = −∞, we can choose ρ ≥ ω Q such that ρ − ω ≤ ε0 and r→∞

1

J(r) ≤ −1 − 2ω Q, ∀r ≥ ρ ω .

(25)

430


Then, we introduce the auxiliary function:

ϕ (t) = [ψ (t)]ω .

(26)

We preliminarily note that, for (almost) every t such that ϕ (t) ≥ ρ , we have

ϕ ′ (t) ≤ −1 − 2ω Q + ω q(t).

(27)

υ

Indeed, for (almost) any fixed t, setting ε = [ϕ (t)]− ω (note that ε ≤ ε0 when ϕ (t) ≥ ρ ), the differential inequality (27) reads 1 ϕ ′ (t) ≤ J([ϕ (t)] ω ) + ω q(t). (28) i) If ϕ (t) ≤ ρ for some t ≥ 0, then ϕ (t + τ ) ≤ 2ρ , for every τ ≥ 0. If not, let τ1 > 0 be such that ϕ (t + τ1 ) > 2ρ , and set τ0 = sup{τ ∈ [0, τ1 ] : ϕ (t + τ ) ≤ ρ }. Integrating (27) on [t + τ0 ,t + τ1 ], we obtain the contradiction 2ρ < ϕ (t + τ1 ) ≤ ρ − (τ1 − τ0 ) − 2ω Q(τ1 − τ0 ) + ω Q(1 + τ1 − τ0 ) < 2ρ . ii) If ϕ (0) > ρ , then ϕ (t∗ ) ≤ ρ , for some t∗ ≤ ϕ (0)(1 + ω Q)−1 . Indeed let t > 0 be such that ϕ (τ ) > ρ for all τ ∈ [0,t]. Integrating (27) on [0,t], we are led to

ρ < ϕ (t) ≤ ϕ (0) − t − 2ω Qt + ω Q(1 + t) ≤ ϕ (0) − (1 + ω Q)t + ρ . Therefore, it must be t < ϕ (0)(1 + ω Q)−1 . 1 1 In order to come back to the original ψ (t), just define R0 = (2ρ ) ω and tR = R ω (1 + ω Q)−1 . By applying the two points discussed above, the proof follows. Remark 3. We get the same result if we suppose that H(r) = l, r→∞ r lim

where l is such that: −ω + ω Kl < 0. Proposition 5. Let α > β ≥ 1 and γ ≥ 0 be such that [0, ∞) satisfying

1 β −1 < , H(t) a continuous function on α −1 1+γ

H(r) = +∞. (29) r Let ψ be a nonnegative absolutely continuous function on [0, ∞) which fulfills, for some K ≥ 0, Q ≥ 0, ε0 > 0 and every ε ∈ (0, ε0 ], the differential inequality lim

r→∞

ψ ′ (t) + ε H(ψ (t)) ≤ K ε α [ψ (t)]β + ε −γ q(t),

(30)


ˆ

t+1

q(y)dy ≤ Q.

(31)

t


ψ (t) ≤ R0 , ∀t ≥ tR , whenever ψ (0) ≤ R. Both R0 and tR can be explicitly computed.

(32)


431

Proof. We can keep the same proof, just take J(r) = −ω r−γυ −υ H(r) + ω Krβ −αυ −γυ . Remark 4. We get the same result if we suppose that the function : r 7→

H(r) r

(33)

is bounded. 1 Proposition 6. Let α > β ≥ 1 and γ ≥ 0 be such that αβ −1 −1 < 1+γ , H(t) a continuous function on [0, ∞) satisfying H(r) lim = +∞. (34) r→∞ r Let ψ be a nonnegative absolutely continuous function on [0, ∞) which fulfills, for some K ≥ 0, Q ≥ 0, ε0 > 0 and every ε ∈ (0, ε0 ], the differential inequality

ψ ′ (t) + ε H(ψ (t)) ≤ K ε α b(t)[ψ (t)]β + ε −γ q(t),

(35)

where q is any nonnegative function satisfying: sup t≥0

ˆ

t+1

q(y)dy ≤ Q,

(36)

t

and b is any function such that b2 satisfy a similar inequality. Then, there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0, such that

ψ (t) ≤ R0 , ∀t ≥ tR ,

(37)

whenever ψ (0) ≤ R. Both R0 and tR can be explicitly computed. Proof. We can write K α 2 ε {b (t) + [ψ (t)]2β } + ε −γ q(t) 2 K K ≤ ε α [ψ (t)]2β + ε −γ {q(t) + ε α +γ b2 (t)} 2 2 K α K α +γ ≤ ε [ψ (t)]2β + ε −γ {q(t) + ε0 b2 (t)}. 2 2 then, it suffices to use the last proposition to obtain the desired estimation.

ψ ′ (t) + ε H(ψ (t)) ≤

1 Proposition 7. Let α > β ≥ 1 and γ ≥ 0 be such that αβ −1 −1 < 1+γ , a(t) a strictly positive continuous function on [0, ∞). Let ψ be a nonnegative absolutely continuous function on [0, ∞) which fulfills, for some K ≥ 0, Q ≥ 0, ε0 > 0 and every ε ∈ (0, ε0 ], the differential inequality

ψ ′ (t) + ε a(t)ψ (t) ≤ K ε α [ψ (t)]β + ε −γ q(t),

(38)


ˆ

t+1

q(y)dy ≤ Q.

(39)

t


ψ (t) ≤ R0 , ∀t ≥ tR , whenever ψ (0) ≤ R. Both R0 and tR can be explicitly computed.

(40)

432


Proof. There exists m > 0 such that a(t) ≥ m for all t ≥ 0. ψ fulfills

ψ ′ (t) + ε mψ (t) ≤ K ε α [ψ (t)]β + ε −γ q(t). We can keep the same proof, just take J(r) = −ω mrω −υ + ω Krβ −αυ −γυ . 5 Uniform bounds to solutions of perturbed systems We consider the time varying differential equation: x˙ = f (t, x) + g(t, x),

(41)

where the origin is supposed to be an equilibrium point globally uniformly exponential stable (GUES) of the nominal system x˙ = f (t, x), that means the existence of a Lyapunov function V (t, x) and constants satisfying: c1 kxk2 ≤ V (t, x) ≤ c2 kxk2 . V˙ (t, x) ≤ −c3 kxk2 . k We have

∂V (t, x)k ≤ c4 kxk. ∂x

∂V ∂V ∂V (t, x) + (t, x) f (t, x) + (t, x)g(t, x), V˙ (t, x) = ∂t ∂x ∂x

(42)

then

∂V V˙ (t, x) ≤ −c3 kxk2 + k (t, x)kkg(t, x)k. ∂x We recall the following definition [10].

(43)

Definition 4. (uniform boundedness) A solution of the perturbed system is said to be globally uniformly bounded if for every α > 0 there exists c = c(α ) such that, for all t0 ≥ 0, kx0 k ≤ α ⇒ kx(t)k ≤ c(α ), Theorem 8. Let λ =

∀ t ≥ t0 .

c3 1 − and q a nonnegative function satisfying c4 2 ˆ t+1 q2 (y)dy ≤ Q. sup t≥0

t

We suppose that kg(t, x)k ≤ + λ )kx(t)k + ε α kx(t)kβ + q(t), then there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0, such that the trajectory x(t) satisfies √ R0 kx(t)k ≤ √ , ∀ t ≥ tR ≥ 0. c1 (−ε cc24

Proof. In this case (43) becomes c2 V˙ (t, x) ≤ −c3 kxk2 + c4 kxk[(−ε + λ )kxk + ε α kxkβ + q] c4 c 2 ≤ −c3 kxk2 + c4 (−ε + λ )kxk2 + c4 ε α kxkβ +1 + c4 kxkq c4


433

c4 c2 + λ )kxk2 + c4 ε α kxkβ +1 + (kxk2 + q2 ) c4 2 c 4 2 ≤ −ε c2 kxk + c4 ε α kxkβ +1 + q2 , 2

≤ −c3 kxk2 + c4 (−ε

or β +1 c4 c4 V˙ (t, x) + ε V (t, x) ≤ β +1 ε α V 2 + q2 . 2 c1 2

Using proposition 2, it comes that there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0, such that V (t, x) ≤ R0 , ∀t ≥ tR . Theorem 9. Let H a continuous positive increasing function on [0, ∞) satisfying lim H(r) = l,

r→∞ √

where l 2 < 2

c1 c4 .

Let λ =

(44)

c3 and q a nonnegative function satisfying c4 sup t≥0

We suppose that kg(t, x)k ≤ (−ε

ˆ

t+1

q(y)dy ≤ Q.

(45)

t

c2 + λ )kx(t)k + ε q(t)H(kx(t)k), c4

(46)

then, there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0, such that the trajectory x(t) satisfies √ R0 kx(t)k ≤ √ , ∀ t ≥ tR ≥ 0. c1 Proof. In this case (43) becomes c2 V˙ (t, x) ≤ −c3 kxk2 + c4 kxk[(−ε + λ )kxk + ε qH(kxk)] c4 c 2 ≤ −c3 kxk2 + c4 (−ε + λ )kxk2 + c4 ε kxkqH(kxk) c4 p c4 1 p V (t, x)) ≤ −ε V (t, x) + √ ε q V (t, x)H( √ c1 c1 1 p c4 ≤ −ε V (t, x) + √ [q2 + ε 2V (t, x)H 2 ( √ V (t, x))], 2 c1 c1 or 1 p c4 c4 V (t, x)) + √ q2 . V˙ (t, x) + ε V (t, x) ≤ ε 2 √ V (t, x)H 2 ( √ 2 c1 c1 2 c1 Using proposition 4, it comes that there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0, such that V (t, x) ≤ R0 , ∀t ≥ tR .

434


Theorem 10. Let λ =

c3 and q a nonnegative function satisfying c4 ˆ t+1 q2 (y)dy ≤ Q. sup t≥0

(47)

t

We suppose that kg(t, x)k ≤ (−ε a(t) + λ )kx(t)k + ε q(t),

(48)

then, there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0, such that the trajectory x(t) satisfies √ R0 kx(t)k ≤ √ , ∀ t ≥ tR ≥ 0. c1

(49)

Proof. In this case (43) becomes V˙ (t, x) ≤ −c3 kxk2 + c4 kxk[(−ε a(t) + λ )kxk + ε q] ≤ −c3 kxk2 + c4 (−ε a(t) + λ )kxk2 + c4 ε kxkq p c4 c4 ≤ −ε a(t)V (t, x) + √ ε q V (t, x), c2 c1 or p c4 c4 V˙ (t, x) + ε a(t)V (t, x) ≤ √ ε q V (t, x) c2 c1 c4 2 c4 ≤ √ ε V (t, x) + √ q2 . 2 c1 2 c1 Using proposition 7, it comes that there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0, such that V (t, x) ≤ R0 ,

∀t ≥ tR .

Theorem 11. Let H a continuous positive increasing function on [0, ∞), we suppose that the function : r 7→ is bounded, let λ =

H(r) r

c3 and q a nonnegative function satisfying c4 ˆ t+1 q2 (y)dy ≤ Q. sup t≥0

(50)

(51)

t

We suppose that kg(t, x)k ≤ −ε H(kx(t)k) + λ kx(t)k + ε α q(t),

(52)

then, there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0, such that the trajectory x(t) satisfies √ R0 kx(t)k ≤ √ , ∀ t ≥ tR ≥ 0. c1

(53)


435

Proof. In this case (43) becomes V˙ (t, x) ≤ −c3 kxk2 + c4 kxk[−ε H(kxk) + λ kxk + ε α q(t)] ≤ −c3 kxk2 − ε c4 kxkH(kxk) + λ c4 kxk2 + c4 ε α kxkq ≤ −ε c4 kxkH(kxk) + c4 ε α kxkq, or

p 1 p c4 c4 √ V˙ (t, x) + ε √ V H( √ V (t, x)) ≤ √ ε α q V (t, x), c2 c2 c1

or p c4 V˙ (t, x) + ε H0 (V (t, x)) ≤ √ ε α q V (t, x) c1 c4 2α c4 ≤ √ ε V (t, x) + √ q2 , 2 c1 2 c1 √ √ where H0 (x) = √cc42 xH( √1c2 x). Using proposition 5, it comes that there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0, such that V (t, x) ≤ R0 , ∀t ≥ tR . 6 Examples A) We consider the nonlinear perturbed system x˙ = f (t, x) + g(t, x)

−x1 . −x2 We consider as Lyapunov function V (t, x) = V (t, (x1 , x2 )) = x21 + x22 , it satisfies where f (t, x)= f (t, (x1 , x2 ))=

∂V kxk2 ≤ V ≤ kxk2 , V˙ ≤ −2kxk2 , k k ≤ 2kxk. ∂x

i) Let g(t, x)=

 3 2  ε x1 (t) + cos2 t 

1 2 (1 − ε )kx(t)k

, ε ∈ [0, 1].

This function satisfies 1 kg(t, x)k ≤ (1 − ε )kx(t)k + ε α kx(t)k2 + cos2 t, 2 using Theorem 8, the solutions are uniformly bounded.   ε q(t) arctankx(t)k , ε ∈ ]0, 1[, q satisfies the conditions of Theorem 9. ii) Let g(t, x)= 1 ε )kx(t)k (1 − 2

This function satisfies

1 kg(t, x)k ≤ (1 − ε )kx(t)k + ε q(t)H(kx(t)k), 2

436


using Theorem 9, the solutions are uniformly bounded. B) We consider the nonlinear differential equation y(t) ˙ + ε y(t) = K ε 3 y2 (t) + sin2 t. Then, there exists R0 > 0 with the following property: for every R ≥ 0, there is tR ≥ 0 such that y(t) ≤ R0 , ∀t ≥ tR ,

(54)

whenever y(0) ≤ R. Both R0 and tR can be explicitly computed.

7 Conclusion In this paper, some new nonlinear integral inequalities with parameter were presented, then these inequalities were used to show that we can prove the boundedness of solutions of perturbed nonlinear time-varying differential equations. The sufficient condition requires the existence of a Lyapunov function for nominal system which can be used to obtain an estimation on the trajectories. References [1] Zelik, S. (2007), Spatially nondecayingsolutions of the 2D Navier-Stokes equation in a strip, Glasg. Math. J., 525-588. [2] Zelik, S. (2008), Weak Spatially non-decaying solutions of the 3D Navier-Stokes equations in cylindrical domains, in: C. Bardos, A. Fursikov(Eds.), instability in models connected with floods flows, in: International Math. Series, vol. 5-6, Springer, New York. [3] Ayels, O. and Penteman, P. (1998), A new asymptotic stability criterion for non linear time varying differential equations, IEEE Trans. Aut. Contr, 43, 968-971. [4] Bay, N.S. and Phat, V.N. (1999), Stability of nonlinear difference time varying systems with delays, Vietnam J. of Math, 4, 129-136. [5] BenAbdallah, A., Ellouze, I., and Hammami, M.A. (2009), Practical stability of nonlinear time-varying cascade systems, J. Dyn. Control Sys., 15, 45-62. [6] Corless, M. (1990), Guaranteed Rates of Exponential Convergence for Uncertain Systems, Journal of Optimization Theory and Applications, 64, 481-494. [7] Corless, M. and Leitmann, G. (1988), Controller Design for Uncertain Systems via Lyapunov Functions, Proceedings of the 1988 American Control Conference, Atlanta, Georgia. [8] Garofalo, F. and Leitmann, G. (1989), Guaranteeing Ultimate Boundedness and Exponential Rate of Convergence for a Class of Nominally Linear Uncertain Systems, Journal of Dynamic Systems, Measurement, and Control, 111, 584-588. [9] Hahn, W. (1967), Stability of Motion, Springer, New York. [10] Khalil, H. (2002), Nonlinear Systems, Prentice Hall. [11] Gronwall, T.H. (1919), Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math., 20(2), 293-296. [12] Bainov, D. and Simenov, P. (1992), Integral inequalities and applications, Kluwer Academic Publishers, Dordrecht. [13] Bellman, R. (1943), The stability of solutions of linear differential equations, Duke Math. J., 10, 643-647. [14] Pata, V. (2011), Uniform estimates of Gronwall types, Journal of Mathematical Analysis and Applications, 264-270. [15] Gatti, S., Pata, V., and Zelik, S. (2009), A Gronwall-type lemma with parameter and dissipative estimates for PDEs, Nonlinear Analysis, 2337-2343. [16] Pata, V. (2011), Uniform estimates of Gronwall types, Journal of Mathematical Analysis and Applications, 264-270. [17] Chepyzhov, V.V., Pata, V., and Vishik, M.I. (2009), Averaging of 2D Navier-Stokes equations with singularly oscilating forces, Nonlinearity, 22, 351-370. [18] Temam, R. (1988), Infinite-Dimensional Systems in Mechanics and Physics, Springer, New York.

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Aims and Scope The interdisciplinary journal publishes original and new research results on applied nonlinear dynamics in science and engineering. The aim of the journal is to stimulate more research interest and attention for nonlinear dynamics and application. The manuscripts in complicated dynamical systems with nonlinearity and chaos are solicited, which includes mathematical theories and methods, physical principles and laws, and computational techniques. The journal provides a place to researchers for the rapid exchange of ideas and techniques in nonlinear dynamics and engineering nonlinearity. Topics of interest include but not limited to • • • • • • • • • • • •

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Continued from inside front cover

Xilin Fu School of Mathematical Science Shandong Normal University Jinan 250014, China Email: [email protected]

C. Nataraj Department of Mechanical Engineering Villanova University, Villanova PA 19085, USA Fax: +1 610 519 7312 Email: [email protected]

Juan J. Trujillo Department de Análisis Matemático Universidad de La Laguna C/Astr. Fco. Sánchez s/n 38271 La Laguna, Tenerife, Spain Fax: +34 922318195 Email: [email protected]

Hamid R. Hamidzadeh Department of Mechanical and Manufacturing Engineering Tennessee State University Nashville, TN37209-1561, USA Email: [email protected]

Raoul R. Nigmatullin Department of Theoretical Physics Kremlevskaiya str.18 Kazan State University, 420008 KAZAN, Tatarstan Russia Email: [email protected]

Yuefang Wang Department of Engineering Mechanics Dalian University of Technology Dalian, Liaoning, 116024, China Email: [email protected]

Reza N. Jazar School of Aerospace, Mechanical and Manufacturing RMIT University Bundoora VIC 3083, Australia Fax: +61 3 9925 6108 Email: [email protected]

Lev Ostrovsky Zel Technology/NOAA ETL Boulder CO, USA 80305 Fax:+1 303 497 7384 Email: [email protected]

Mikhail V. Zakrzhevsky Institute of Mechanics Riga Technical University 1 Kalku Str., Riga LV-1658 Latvia Email: [email protected]

Stefano Lenci Dipartimento di Ingegneria Civile Edile e Architettura, Universita' Politecnica delle Marche via Brecce Bianche, 60131 ANCONA, Italy Fax: +39 071 2204576 Email: [email protected]

Richard H. Rand Department of Mathematics Cornell University Ithaca, NY 14853-1503, USA Fax:+1 607 255 7149 Email: [email protected]

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Indexed by Scopus and zbMATH

Journal of Applied Nonlinear Dynamics Volume 7, Issue 4

December 2018

Contents Switching Tracking Control and Synchronization of Four-Scroll Hyperchaotic Systems K. S. Ojo, A. B. Adeloye, A. O. Busari….………....……………………..……......

329－335

Hopf Bifurcation and Stability Analysis of a Predator-Prey System with Holling Type IV Functional Response Z. Lajmiri, R. Khoshsiar Ghaziani, I. Orak…………...…………………………..

337－348

Delay-Coupled Mathieu Equations in Synchrotron Dynamics Revisited: Delay Terms in the Slow Flow Alexander Bernstein, Richard Rand.........................................................................

349－360

A Computational and Theoretical Review on the Motion of a Spinning Spherical Particle in Media with Different Viscosities F. L. Braga, I. G. Pauli, V. S. Amorim...………...….............................………..…

361－369

Existence Result for a Neutral Fractional Integro-Differential Equation with State Dependent Delay K. Jothimani, N. Valliammal, C. Ravichandran................................................…

371－381

Synchronization of Unified Chaotic System via Output Feedback Control Scheme Xue-Rong Tao, Ling Tang, Ping He…………………………………………….....

383－392

A Novel Quasigroup Substitution Scheme for Chaos Based Image Encryption Vinod Patidar, N. K. Pareek, G. Purohit…………………………………...…......

393－412

On Some Properties of Memristive Lorenz Equation – Theory and Experiment P. Saha, D. C. Saha, A. Ray, A. Roy Chowdhury……..…......................…….…...

413－423

Nonlinear Integral Inequalities WITH Parameter and Applications Taoufik Ghrissi, M. A. Hammami…………………………………………………

425－436

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Journal of Applied Nonlinear Dynamics

Journal of Applied Nonlinear Dynamics

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