Improved Time Response of Stabilization in Synchronization of ... - MDPI

systems Article

Improved Time Response of Stabilization in Synchronization of Chaotic Oscillators Using Mathematica Mohammad Shahzad 1, *, Israr Ahmad 1,2 , Azizan Bin Saaban 2 and Adyda Binti Ibrahim 2 1 2

*

College of Applied Sciences, Ministry of Higher Education, Nizwa 611, Sultanate of Oman; [email protected] School of Quantitative Sciences, College of Arts & Sciences, UUM, Keddha 06010, Malaysia; [email protected] (A.B.S.); [email protected] (A.B.I.) Correspondence: [email protected]; Tel.: +968-9958-4197

Academic Editor: Ockie Bosch Received: 27 December 2015; Accepted: 15 June 2016; Published: 22 June 2016

Abstract: Chaotic dynamics are an interesting topic in nonlinear science that has been intensively studied during the last three decades due to its wide availability. Motivated by much researches on synchronization, the authors of this study have improved the time response of stabilization when parametrically excited Φ6 —Van der Pol Oscillator (VDPO) and Φ6 —Duffing Oscillator (DO) are synchronized identically as well as non-identically (with each other) using the Linear Active Control (LAC) technique using Mathematica. Furthermore, the authors have synchronized the same pairs of the oscillators using a more robust synchronization with faster time response of stability called Robust Adaptive Sliding Mode Control (RASMC). A comparative study has been done between the previous results of Njah’s work and our results based on Mathematica via LAC. The time response of stabilization of synchronization using RASMC has been discussed. Keywords: chaos synchronization; LAC; Mathematica; chaotic oscillators; RASMC

1. Introduction The study of chaotic behavior in nonlinear systems has attracted much attention because of many possible applications in various fields of science and technology. Most of the research has been devoted to the modeling of new chaotic systems together with the control and synchronization [1]. Thus far much work based on modeling, as well as various new control and synchronization techniques, has been carried out and is worth citing. For example, the sliding mode control [2–5], adaptive control [6–8], linear active control [9–13], linear feedback control [14–16], projective synchronization [17–19], nonlinear active control [20,21] and backstepping control [22], to mention but a few. A recent study of Shahzad [23] focused the attention of researchers on how to choose a model based synchronization technique and the appropriate mathematical tools for simulation. Pourmahmood et al. [24] have developed a Robust Adaptive Sliding Mode Control (RASMC) and implemented it successfully on three well known chaotic systems (Lorenz, Chen and Liu) using MATLAB for all of the simulations, but when the same study was done using Mathematica by Shahzad [23], remarkable changes were observed in terms of time response to stabilize the synchronization. Motivated by the aforementioned studies, we synchronize the identical and non-identical pairs 6 of Φ —VDPO and DO using the LAC and RASMC, respectively. However, the synchronization of Φ6 —VDPO and DO using the LAC has already been done by Njah [12] but when the same work of Njah [12] was repeated via LAC and using Mathematica, remarkable changes had been found in the time response of stabilization of synchronization. On the other hand, the faster time

Systems 2016, 4, 25; doi:10.3390/systems4020025

www.mdpi.com/journal/systems

Systems 2016, 4, 25

2 of 21

response of stabilization of synchronization performances of the RASMC forced us to implement it on the same pairs of systems studied by Njah [12]. To the best of our knowledge, this kind of study has never been done before. The main objective behind the implementation of RASMC is that the sliding mode control is one of the robust control methods and has many interesting features such as low sensitivity to external disturbances and robustness to the plant uncertainties due to structural variations and un-modeled dynamics. The sliding mode controller is composed of an equivalent control part that describes the behavior of the system when the trajectories stay over the sliding surface and a variable structure control part that enforces the trajectories to reach the sliding surface and remain on it evermore. The adaptive control is a suitable approach to overcome system uncertainties, especially uncertainties derived from uncertain parameters. The adaptive sliding mode control has the advantages of combining the robustness of the sliding mode control with the tracking facilities of the adaptive control ([24], and the references therein). The rest of the paper has been organized as follows: In Section 2, the three identical pairs 6 of Φ —VDPO and DO, respectively, and non-identical pairs of Φ6 —VDPO and DO have been synchronized using the LAC. Section 3 is devoted to the brief description of RASMC as well as its implementation on identical synchronization of Φ6 —VDPO and DO, respectively, and non-identical pair of Φ6 —VDPO and DO. Lastly, the whole study has been concluded in Section 4. 2. Synchronization Using LAC In this section, we synchronize the identical and non-identical pairs of Φ6 —VDPO and DO, respectively, using the LAC technique for chaos synchronization that was proposed by Bai and Lonngren [9] and it has recently been accepted as one of the most efficient techniques for synchronizing both identical and non-identical chaotic systems because of its simple implementation in practical systems [25–28]. It can be easily designed according to the given conditions of the chaotic system as a way of accomplishing synchronization globally asymptotically, if the nonlinearity of the system is known. There are no derivatives in the controller and no need to calculate the Lyapunov exponents to execute the controller. These characteristics give an advantage to the technique over other conventional synchronization techniques. 2.1. Description of the Models Since the chaotic systems are very complex nonlinear systems, they are exceptionally sensitive to tiny changes in their initial conditions and parameters variations. With the passage of time and due to the potential applications of chaotic systems in certain scientific fields, many chaotic and hyperchaotic systems have been investigated (Lorenz, Chen and Liu, etc.). In this direction, the Φ6 —VDPO and DO are periodically self-excited and have rich applications in various disciplines like electronics, physics, engineering, neurology and biological sciences [29–31]. Njah [12] studied and investigated the synchronizations of Φ6 —VDPO and DO with applications to secure communication using the LAC technique based on the Lyapunov stability theory and the Routh-Hurwitz criterion. The synchronization schemes have been studied without considering the external disturbances and model uncertainties. However, in practical applications, either environmental changes (noise) may occur any time or lack of parameters knowledge may disturb the stability of the synchronized system and this uncertainty or environmental noise cannot be simply ignored. The Φ6 —VDPO and DO are classical examples of self-oscillatory and periodic systems and are now considered as very valuable mathematical models that can be utilized in much more complex and modified systems. In these models, there exist two frequencies, namely periodic forcing and self-oscillations. The energy is generated at low amplitudes and dissipated at high amplitude. The dynamics of chaotic parametrically excited Φ6 —VDPO [12] are given by the following mathematical model: Φ6 ´ VDPO :

..

.

x ´ µ1 p1 ´ x2 qx ` α1 t1 ` η1 cosp2ω1 tqu x ` β1 x3 ` λ1 x5 “ f 1 cosω1 t

(1)

Systems 2016, 4, 25

3 of 21

The dynamics of another chaotic parametrically excited Φ6 —DO [12] are given by the following mathematical model: ..

.

Φ6 ´ DO : x ` µ2 x ` α2 t1 ` η2 cosp2ω2 tqu x ` β2 x3 ` λ2 x5 “ f 2 cosω2 t

(2)

where the Φ6 —VDPO and DO exhibit chaotic attractors for the following parameter values: µ1 “ 0.4, α “ 1.0, β “ ´0.7, λ1 “ 0.1, η1 “ 0.7, f 1 “ 9, ω1 “ 3.14 and µ2 “ 0.4, α2 “ 0.46, β2 “ 1, λ2 “ 0.1, f 2 “ 4.5, η2 “ 0.7, ω2 “ 0.86, respectively [12]. 2.2. Synchronization of Two Identical Φ6 —VDPO Oscillators via LAC To achieve synchronization between two identical Φ6 —VDPO, let us consider the master–slave systems synchronization scheme for two coupled identical chaotic Φ6 —VDPO that can be written by . . choosing x “ x1 and x1 “ x2 and x “ y1 and x “ y2 in Equation (1) for master and slave systems, respectively, as follows: # Master System : # Slave System :

.

x 1 “ x2 , . x2 “ µ1 p1 ´ x12 qx2 ´ α1 t1 ` η1 cosp2ω1 tqu x1 ´ β1 x13 ´ λ1 x15 ` f 1 cosω1 t

(3)

.

y1 “ y2 ` u1 ptq, . y2 “ µ1 p1 ´ y21 qy2 ´ α1 t1 ` η1 cosp2ω1 tqu y1 ´ β1 y31 ´ λ1 y51 ` f 1 cosω1 t ` u2 ptq

(4) where rx1 ptq, x2 ptqs and ry1 ptq, y2 ptqs P are the state variables of master and slave systems, respectively; µ1 , α1 , β1 , λ1 , f 1 , η1 and ω1 are the parameters involved in Equations (3) and (4) and uptq “ ru1 ptq, u2 ptqsT P R2ˆ1 are the control inputs yet to be determined. Now, the error dynamics (ei “ yi ´ xi , for i “ 1, 2) from Equations (3) and (4) can be written as follows: T

T

R2

.

e1 “ e2 ` u1 ptq . e2 “ µ1 e1 ´ µ1 py21 y2 ´ x12 x2 q ´ α1 t1 ` η1 cosp2ω1 tqu e1 ´ β1 py31 ´ x13 q ´ λ1 py51 ´ x15 q ` u2 ptq

(5)

In order to make the error dynamics linear, let us redefine u1 & u2 as follows: u1 ptq “ v1 ptq u2 ptq “ µ1 py21 y2 ´ x12 x2 q ` β1 py31 ´ x13 q ` λ1 py51 ´ x15 q ` v2 ptq

(6)

6 Now the linear error dynamical system can be written as: .

e1 “ e2 ` v1 ptq . e2 “ µ1 e1 ´ α1 t1 ` η1 cosp2ω1 tqu e1 ` v2 ptq

(7)

The linear by v1 pe1 , e2 q and v2 pe1 , e2 q that are ˜ error ¸ dynamics ˜ ¸(Equation (7))˜is controlled ¸ v1 e1 a b defined as: “ D where D “ is a constant feedback matrix yet to be v2 e2 c d ˜ . ¸ ˜ ¸ e1 e1 determined and the error dynamics (Equation (7)) can be written as: “ C where . e2 e2 ˜ ¸ a 1`b C “ is the coefficient matrix. According to the c ´ α1 t1 ` η1 cosp2ω1 tqu µ1 ` d Lyapunov stability theory and Routh–Hurwitz criteria, choose a ` d ` µ1 ă 0 and tc ´ α1 p1 ` η1 cos2ω1 tqu p1 ` bq ´ apµ1 ` dq ă 0 for the stabilization of the synchronization of Equations (3) and (4).

Systems 2016, 4, 25

4 of 21

Let a ` d ` µ1 “ tc ´ α1 p1 ` η1 cos2ω1 tqu p1 ` bq ´ apµ1 ` dq “ ´E for E ą 0. We choose a “ b “ 0 and E “ 1, which yields u1 ptq “ 0 and u2 ptq “ µ1 py21 y2 ´ x12 x2 q ` β1 py31 ´ x13 q ` λ1 py51 ´ x15 q ` tα1 p1 ` η1 cos2ω1 tq ´ Eu e1 ´ pµ1 ` Eqe2 . For the same values of parameters (µ1 “ 0.4, α1 “ 1, β1 “ ´0.7, λ1 “ 0.1, f 1 “ 9, η1 “ 0.7, ω1 “ 3.14 and E “ 1) and initial conditions (x1 p0q “ 0.1; x2 p0q “ 0.2; y1 p0q “ 2.2; y2 p0q “ 0.05) taken by [12], we have repeated all simulations using Mathematica. The following are the graphs of synchronization of two identical Φ6 —VDPO oscillators: 2.3. Synchronization for Two Identical Φ6 —DO via LAC To achieve synchronization between two identical Φ6 —DO, let us consider the master–slave systems synchronization scheme for two coupled identical chaotic Φ6 —DO that can be written by . . taking x “ x1 and x1 “ x2 and x “ y1 and x “ y2 in Equation (2) for master and slave systems, respectively, as follows: # Master System : # Slave System :

.

x 1 “ x2 , . x2 “ ´µ2 x2 ´ α2 t1 ` η2 cosp2ω2 tqu x1 ´ β2 x13 ´ λ2 x15 ` f 2 cosω2 t

(8)

.

y1 “ y2 ` u1 ptq, . y2 “ ´µ2 y2 ´ α2 t1 ` η2 cosp2ω2 tqu y1 ´ β2 y31 ´ λ2 y51 ` f 2 cosω2 t ` u2 ptq

(9)

where rx1 ptq, x2 ptqsT and ry1 ptq, y2 ptqsT P R2 are the state variables; µ2 , α2 , β2 , λ2 , f 2 , η2 and ω2 are the parameters involved in Equations (8) and (9); and uptq “ ru1 ptq, u2 ptqsT P R2ˆ1 are the control inputs yet to be determined. Using LAC technique as in Section 2.2, someone can find the controllers u1 ptq “ 0 and u2 ptq “ β2 py31 ´ x13 q ` λ2 py51 ´ x15 q ` tα2 p1 ` η2 cos2ω2 tq ´ Eu e1 ` pµ2 ´ Eqe2 . For the same values of parameters (µ2 “ 0.4, α2 “ 0.46, β2 “ 1, λ2 “ 0.1, f 2 “ 4.5, η2 “ 0.7, ω2 “ 0.86 and E “ 1) and initial conditions (x1 p0q “ 0; x2 p0q “ 1.5; y1 p0q “ 0.5; y2 p0q “ 1) as taken by Njah [12], we have repeated all simulations using Mathematica. The following are the graphs of synchronization of two identical Φ6 —DO: 2.4. Synchronization for Φ6 —VDPO and DO via LAC To achieve synchronization between Φ6 —VDPO and DO, let us consider the master-slave systems synchronization scheme for two coupled Φ6 —VDPO and DO that can be written by taking x “ x1 . . and x1 “ x2 and x “ y1 and x “ y2 in Equations (1) and (2) for master and slave systems, respectively, as follows: # . x 1 “ x2 , Master System : (10) . x2 “ µ1 p1 ´ x12 qx2 ´ α1 t1 ` η1 cosp2ω1 tqu x1 ´ β1 x13 ´ λ1 x15 ` f 1 cosω1 t # Slave System :

.

y1 “ y2 ` u1 ptq, . y2 “ ´µ2 y2 ´ α2 t1 ` η2 cosp2ω2 tqu y1 ´ β2 y31 ´ λ2 y51 ` f 2 cosω2 t ` u2 ptq

(11)

where rx1 ptq, x2 ptqsT and ry1 ptq, y2 ptqsT P R2 are the state variables of master and slave systems; µ1 , α1 , β1 , λ1 , f 1 , η1 , ω1 , µ2 , α2 , β2 , λ2 , f 2 , η2 and ω2 are the parameters involved in Equations (10) and (11) and uptq “ ru1 ptq, u2 ptqsT P R2ˆ1 are the control inputs yet to be determined. Using LAC technique, as it has been implemented in the last two subsections, someone can find the controllers: u1 ptq “ 0 u2 ptq “ pµ1 ` µ2 qx2 ´ µ1 x12 x2 ` α2 p1 ` η2 cos2ω2 tqx1 ´ α1 p1 ` η1 cos2ω1 tqx1 ` β2 y31 ´ β1 x13 `λ2 y51 ´ λ1 x15 ´ f 2 cosω2 t ` f 1 cosω1 t ` tα2 p1 ` η2 cos2ω2 tq ´ Eu e1 ` pµ2 ´ Eqe2

(12)

u1 (t )  0 u2 (t )  (1  2 ) x2  1 x12 x2   2 (1  2 cos 22t ) x1  1 (1  1 cos 21t ) x1  2 y13  1 x13

(12)

  2 y15  1 x15  f 2 cos 2t  f1 cos 1t   2 (1  2 cos 22t )  E e1  ( 2  E )e2 Systems 2016, 4, 25

For the same values of parameters ( 1 = 0.4 , 1  1 , 1  0.7 , 1  0.1 , f1 = 9 , 1 = 0.7 , 5 of 21

1  3.14 , 2 = 0.4 , 2  0.46 , 2  1 ,  2  0.1, f 2 = 4.5 , 2 = 0.7 , 2  0.86 and E  1 ) For the same values of parameters (µ “ 0.4, α1 “ 1, β1 “ ´0.7, λ1 “ 0.1, f 1 “ 9, η1 “ 0.7, and initial conditions ( x1 (0)  0.1 ; x2 (0) 1 0.2 ; y1 (0)  0 ; y2 (0)  1.5 ) as taken by Njah [12], we

ω1 “ 3.14, µ2 “ 0.4, α2 “ 0.46, β2 “ 1, λ2 “ 0.1, f 2 “ 4.5, η2 “ 0.7, ω2 “ 0.86 and E “ 1) and initial have repeated all “ simulations using The“following are the graphs of synchronization of conditions (x1 p0q 0.1; x2 p0q “ 0.2; Mathematica. y1 p0q “ 0; y2 p0q 1.5) as taken by Njah [12], we have repeated 6—VDPO and DO. Φ 6 all simulations using Mathematica. The following are the graphs of synchronization of Φ —VDPO and DO. 2.5. Results and Discussions

2.5. Results and Discussions In Sections 2.2–2.4, three different pairs of Φ6—VDPO and DO have been synchronized using 6 —VDPO LACIn technique. our study, simulations based on Mathematica that provide us the remarkable Sections In 2.2–2.4, threeall different pairsare of Φ and DO have been synchronized using changes in the time of stabilization of synchronization. Earlier, in the same study of Njah [12], for LAC technique. In our study, all simulations are based on Mathematica that provide us the remarkable 6—VDPO and non-identical pairs (i.e., Φ6—VDPO and DO), controllers were identical pairs of Φ changes in the time of stabilization of synchronization. Earlier, in the same study of Njah [12], for t  60 and activated pairs at around identical pairs Φ6—DO, the controllers activated at identical of Φ6 —VDPO andfor non-identical pairsof(i.e., Φ6 —VDPO and DO),were controllers were 6 t  100 around . On the other hand, for the same pairs and same technique (LAC) if Mathematica is activated at around t “ 60 and for identical pairs of Φ —DO, the controllers were activated at around used, observe thesame remarkable changes the time (LAC) of stabilization (Figures 1–13). tbeing “ 100. Onsomeone the othercan hand, for the pairs and same in technique if Mathematica is being t  41–13). In our study, can for observe all of the not only do in controllers around but In secure used, someone the cases, remarkable changes the time ofactivate stabilization (Figures our communication scheme (Figures 3, controllers 4, 7, 8, 11activate and 12), convergence of errors defined by study, for all of the cases, not only do around t “ 4 butbsecure communication

scheme 11 and13) 12),also convergence of errorsatdefined by eptq “ when e12 ptqsimulation ` e22 ptq (Figure 13) (Figure start to stabilize the same time is done e(t )  (Figures e12 (t )  3, e224,(t )7, 8, also start to stabilize at the same time when simulation is done using Mathematica. Furthermore, it using Mathematica. Furthermore, it may also be observed that the error states converged to the origin may also be observed that the error states converged to the origin in the range of [´0.5, 1.5] very in the range of [−0.5, 1.5] very smoothly and quickly as compared to the work done by Njah [12]. smoothly and quickly as compared to the work done by Njah [12]. These features give advantages to These features give advantages to the current study. the current study. 2.0

e1

1.5

e2

1.0 0.5 0.0 0.5 1.0 1.5 0 Systems 2016, 4, 25

2

4 6 8 Fig 1: Time Series of e1 , e2 Figure 1. Time Series of e1 & e2 . Figure 1. Time Series of e1 & e2 .

10 6 of 22

3 2 1 0 x1

1

y1

2

x2

3

y2

0

2

1 2 3 4 Fig 2: Time Series of x1 , y1 , x2 & y2 Figure Figure 2. 2. Time Time Series Series of of x1x1, ,yy11, ,xx22 & & yy22..

5

x1 s

1

0

2 3 3 0 0

1 2 3 4 1 2: Time 2Series of x13, y1 , x2 & y42 Fig Figure 2. Time of xx11, ,yy11,, xx22 & & y22 . Fig 2: Time Series of Figure 2. Time Series of x1 , y1 , x2 & y2 .

Systems 2016, 4, 25

x2 y2 y2

5 5 6 of 21

x1 x1 s s

2 2 1 1 0 0 1 1 0 0

2 4 6 8 2 Fig 3: Time 4 Series of 6 x1 & s 8 Figure & ss . Fig 3. 3: Time Time Series Series of xx11 & Figure Time Series Figure 3. 3. Time Series of of xx11 & & ss..

10 10

0.0 0.0 0.5 0.5 1.0 1.0 1.5 1.5

m m m' m'

2.0 2.0 0 0

Systems 2016, 4, 25

2 4 6 8 2 Fig 4: Time 4 Series of 6 m & m' 8 1 Figure Time mm&&m Fig4. TimeSeries Seriesof m''. . &m Figure 4.4:Time Series ofof m Figure 4. Time Series of m & m ' .

10 10

e1

0.4

e2

0.2 0.0 0.2 0.4 0

2

4 6 8 Fig 5: Time Series of e1 , e2 Figure 5. Time Series of e & e . Figure 5. Time Series of e11 & e22 .

10

3 x1

2

y1 x2

1

y2

0 1 2 3 0

1 2 3 4 Fig 6: Time Series of x1 , y1 , x2 & y2 Figure 6. Time Series of x1 , y1 , x2 & y2 .

5

7 of 22

0.4 0.4 0 0 Systems 2016, 4, 25

3 3 2 2 1 1 0 0 1 1 2 2 3 3 0 0

2 2

4 6 8 4 6 8 Fig 5: Time Series of e1 , e2 Time Series FigureFig 5. 5: Time Series ofofe1e1&, ee22 . Figure 5. Time Series of e1 & e2 .

10 10 7 of 21

x1 x1 y1 y1 x2 x2 y2 y2

1 2 3 4 1 6: Time 2Series of x13, y1 , x2 & y42 Fig Figure Fig 6. 6:Time Time Series Seriesof of xx11,, yy1 , x22 & & yy22 . Figure 6. 6. Time Time Series Series of of xx1 ,,yy1 ,,xx2 & & yy2 .. Figure 1

1

2

5 5

2

2 2

x1 x1 s s

1 1 0 0 1 1

Systems 2016, 4, 25

2 2 0 0

2 4 6 8 2 Fig 7: Time 4 Series of 6 x1 & s 8 Figure 7. Time Series of & ss.. Figure Fig 7. 7:Time Time Series Seriesof of xxx11 & Figure 7. Time Series of x1 & s .

10 10

0.1 0.0 0.1 0.2 0.3 0.4

m m'

0.5 0

2

4 6 8 Fig 8: Time Series of m & m' Figure 8. Time Series of m & m1 . Figure 8. Time Series of m & m ' .

10

e1 e2

1.0

0.5

0.0 0

2

4 6 8 Fig 9: Time Series of e1 & e2 Figure 9. Time Series of e1 & e2 .

10

8 of 22

0.4 0.5 0.5 0 0 Systems 2016, 4, 25

m m' m'

2 4 6 8 2 Fig 8: Time 4 Series of 6 m & m' 8 Fig8.8:Time TimeSeries Seriesofof m m& &m m'' . Figure Figure 8. Time Series of m & m ' .

10 10 8 of 21

e1 e1 e2 e2

1.0 1.0

0.5 0.5

0.0 0.0 0 0

Systems 2016, 4, 25

2 4 6 8 2 Fig 9: Time 4 Series of 6 e &e 8 1 2 & ee22 . Figure Fig 9. 9: Time Time Series Seriesof ee11 & Figure & ee22.. Figure 9. 9. Time Time Series Series of of e1e1 &

3 3 2 2 1 1 0 0 1 1 2 2 3 3 0 0

1 2 3 4 1 10: Time2Series of x3, y , x & 4y Fig 1 1 2 2 Figure 2 & 11,,,xx Fig 10. 10: Time Series Series of of & y22.. Figure 10. Time Time of xx1x1,1, ,yyy 1 x22 & y2 Figure 10. Time Series of x1 , y1 , x2 & y2 .

10 10

x1 x1 y1 y1 x2 x2 y2 y2

5 5

x1

2

s

1

0

1

0

2

4 6 8 Fig 11: Time Series of x1 & s Figure Figure 11. 11. Time Time Series Series of of x1x1 & & s. s.

10

0.1 0.0 0.1 0.2 m

0.3

m'

0

2

4 6 8 Fig 12: Time Series of m & m' Figure 12. Time Series of m & m ' .

10

9 of 22

1 0 0

2 4 6 8 2 Fig 11: Time 4 6 8 Series of x1 & s Fig 11: Time Series of Figure 11. Time of xx11& &ss . Figure 11. Time Series of x1 & s .

Systems 2016, 4, 25

10 10 9 of 21

0.1 0.1 0.0 0.0 0.1 0.1 0.2 0.2 m m m' m'

0.3 0.3 0 0

2 4 6 8 2 Fig 12: Time 4 Series6of m & m' 8 Fig 12:Time TimeSeries Seriesofof m m& &m m' Figure Figure 12. 12. Time Series of m & m1'.. Figure 12. Time Series of m & m ' .

e for e for e for e for e for e for

2.0 2.0 1.5 1.5

10 10

6 VP & DO 6 VP & DO 6 VP 6 VP 6 DO 6 DO

1.0 1.0 0.5 0.5 0.0 0.0

0 0

2 4 6 2 Fig 13: Time 4 Series of e 6 Figure 13. Time of Fig 13: Figure 13. Time Time Series Series of of ee.e. Figure 13. Time Series of e.

8 8

3. SynchronizationUsing Using RASMC RASMC 3. Synchronization 3. Synchronization Using RASMC 6—VDPO In section,we wesynchronize synchronizethe the identical non-identical pairs Φ6 —VDPO andusing DO In this section, identical andand non-identical pairs of Φof and DO 6—VDPO and DO using In this section, we synchronize the identical and non-identical pairs of Φ using RASMC technique a very quick response stabilizingthe thesynchronization synchronization of chaotic RASMC technique that that has ahas very quick response in instabilizing chaotic RASMC technique thatwe has a verythe quick response in stabilizing the synchronization of chaotic systems Below describe technique in systems [32–36]. [32–36]. describe the technique in details. details. systems [32–36]. Below we describe the technique in details. 3.1. 3.1. Description Description of of RASMC RASMC 3.1. Description of RASMC For For the the n-dimensional n-dimensional master master and and slave slave systems systems with with external external uncertainties, uncertainties, disturbances disturbances and and For the n-dimensional master and slave systems with external uncertainties, disturbances and unknown unknown parameters, parameters, the the RASMC RASMC [24] [24] is is described described as as follows: follows: unknown parameters, the RASMC [24] is described as follows: . Master system : xptq “ fpxq ` Fpxqθ ` ∆fpx, tq ` dm ptq (13) .

Slave System : yptq “ gpyq ` Gpyqψ ` ∆gpy, tq ` ds ptq ` uptq

(14)

where xptq “ rx1 , x2 , . . . , xn sT are the state vectors, fpxq “ r f 1 pxq, f 2 pxq, . . . , f n pxqsT are the continuous nonlinear functions, Fi pxq, i “ 1, 2, . . . , n, is ith row of an n ˆ n matrix pFpxqq whose elements are continuous nonlinear functions, θ “ rθ1 , θ2 , . . . , θn sT are the unknown vector parameters, “ ‰T and ∆fpx, tq “ r∆ f 1 px, tq, ∆ f 2 px, tq, . . . , ∆ f n px, tqsT and dm ptq “ d1m ptq, d2m ptq, . . . , dm are the n ptq vectors of unknown uncertainties and external disturbances of the master system, respectively. yptq “ ry1 , y2 , . . . , yn sT are the state vectors, gpyq “ rg1 pyq, g2 pyq, . . . , gn pyqsT are the continuous nonlinear functions, Gi pyq, i “ 1, 2, . . . , n, is ith row of an n ˆ n matrix pGpyqq whose elements are continuous nonlinear functions, ψ “ rψ1 , ψ2 , . . . , ψn sT are the unknown vector parameters, “ ‰T ∆gpy, tq “ r∆g1 py, tq, ∆g2 py, tq, . . . , ∆gn py, tqsT and ds ptq “ d1s ptq, d2s ptq, . . . , dsn ptq are the vectors

Systems 2016, 4, 25

10 of 21

of unknown uncertainties and external disturbances of the slave system, respectively, and uptq “ ru1 ptq, u2 ptq, . . . , un ptqsT is the vector of control inputs. Assumption 1: Since the trajectories of chaotic systems are always bounded, then the unknown uncertainties ∆fpx, tq and ∆gpy, tq are assumed to be bounded. Therefore, there exist appropriate positive constants αim and αis , i “ 1, 2, . . . , n such that |∆ f i px, tq| ă αim

and |∆gi py, tq| ă αis ,

ñ |∆ f i px, tq ´ ∆gi py, tq| ă αi ,

i “ 1, 2, . . . n

i “ 1, 2, . . . , n, where αi are unknown constants

(15) (16)

Assumption 2: In general, it is assumed that the external disturbances are norm-bounded in C1 , i.e., |dim ptq| ă βim ñ |dim ptq ´ dis ptq| ă βi ,

and |dis ptq| ă βis ,

i “ 1, 2, . . . , n

(17)

i “ 1, 2, . . . , n, where βi are unknown constants

(18)

To solve the synchronization problem, the error between the master system (Equation (13)) and slave systems (Equation (14)) can be defined as eptq “ xptq ´ yptq. Then, from Equations (13) and (14), the error dynamics can be written as: .

eptq “ fpxq ` Fpxqθ ` ∆fpx, tq ` dm ptq ´ gpyq ´ Gpyqψ ´ ∆fpy, tq ´ ds ptq ´ uptq

(19)

It is clear that the synchronization problem can be transformed to the equivalent problem of stabilizing the error system (Equation (19)). The objective of this paper is to show that for any given master chaotic system (Equation (13)) and slave chaotic system (Equation (14)) with the uncertainties, external disturbances and unknown parameters a suitable feedback control law uptq is designed such that the asymptotical stability of the resulting error system (Equation (19)) can be achieved in the sense that lim |xptq ´ yptq| Ñ 0 is for the systems under consideration. tÑ8

Let us consider now the appropriate sliding surface with the desired behavior. Therefore, the sliding surface suitable for the technique can be designed as: si ptq “ λi ei ptq,

i “ 1, 2, . . . , n

(20)

where si ptq P R psptq “ rs1 ptq, s2 ptq, . . . , sn ptqsq and the sliding surface parameters λi are positive constants. After designing the suitable sliding surface, let us determine the input control signal uptq to guarantee that the error system trajectories reach to the sliding surface sptq “ 0 (i.e., to satisfy the . reaching condition sptqsptq ă 0) and stay on it permanently. Therefore, to ensure the existence of the sliding motion a discontinuous control law (with minimum chattering effect) is proposed as: ` ˘ ˆ i ` αˆ i ` βˆ i sgnpsi q ` k i tanhpεsi q, ui ptq “ f i pxq ´ gi pyq ` Fi pxqθˆ i ´ Gi pyqψ

for i “ 1, 2, . . . , n

(21)

ˆ i , αˆ i , and βˆ i are estimations for θi , ψi , and αi , respectively, k i ą 0, i “ 1, 2, . . . , n are the where θˆ i , ψ switching gain constant, and ε ą 0. To tackle the uncertainties, external disturbances and unknown parameters, appropriate update laws are defined as: . ˆ θˆ “ rFpxqsT γ, θp0q “ θˆ 0 . ˆ “ ´ rGpyqsT γ, ˆ ˆ0 (22) ψ ψp0q “ψ .

.

αˆ i “ βˆ i “ λi |si | ,

αˆ i p0q “ αˆ i0 & βˆ i p0q “ βˆ i0

ˆ 0 , αˆ i0 and βˆ i0 are the initial values of the update parameters where γ “ rλ1 s1 , λ2 s2 , . . . , λn sn sT and θˆ 0 , ψ ˆ ψ, ˆ αˆ i and βˆ i , respectively. θ,

Systems 2016, 4, 25

11 of 21

Based on the control input in Equation (21) and update laws in Equation (22) as used to guarantee . the reaching condition sptqsptq ă 0 and to ensure the occurrence of the sliding motion, we have the following theorem. Theorem 1: Consider the error dynamics in Equation (19), this system is controlled by uptq in Equation (13) with update laws in Equation (14). Then the error system trajectories will converge to the sliding surface sptq “ 0. In this regard, we consider a Lyapunov function (that is a positive definite function also) as follow: n

Vptq “

˘2 ı 1 ` 1 1ÿ” 2 2 ˆ ´ ψ||2 ` ||θˆ ´ θ|| ` ||ψ si ` pαˆ i ´ αi q2 ` βˆ i ´ βi 2 2 2

(23)

i“1

In order to apply the RASMC to synchronize the identical pairs of chaotic Φ6 —VDPO; Φ6 —DO and non-identical pair Φ6 —VDPO and DO, external uncertainty for master and slave systems have been chosen as: ∆ f i pxi , tq Ñ 0.5sinxi and ∆gi pyi , tq Ñ ´0.5sinyi , respectively; external disturbance for master and slave system: dim ptq Ñ 0.1sint and dis ptq Ñ ´0.1sint for all i “ 1, 2, respectively; ε “ 100; ˆ i p0q “ 2, αˆ i p0q “ 3 and βˆ i p0q “ 4; and for secure initial values of update parameters θˆi p0q “ 1, ψ communication message signal pmq “ 0.05sin2t. 3.2. Synchronization of Two Identical Φ6 —VDPO Using RASMC In this section, we synchronize the identical pairs of chaotic Φ6 —VDPO using RASMC under the effect of external uncertainty and external disturbance for both master and slave systems. After adding the external uncertainty and disturbances, Equation (1) can be written as a pair of master and slave systems: « ff « ff« ff « ff « ff . x2 0 1 0.5sinx1 0.1sint 0 ` ` (24) x“ ` 0 x2 µ1 0.5sinx2 0.1sint A loomoon loooooomoooooon looomooon looooooomooooooon loooooomoooooon fpxq

«

ff

«

Fpx,tq

ff«

dm ptq

∆fpx,tq

θ

ff

«

ff

«

ff « ff 0 y2 0 1 ´0.5siny1 ´0.1sint u1 ptq y“ ` ` ` ` B 0 y2 µ1 ´0.5siny2 ´0.1sint u2 ptq loomoon loooooomoooooonlooomooon loooooooomoooooooon looooooomooooooon loooomoooon .

gpyq

Gpy,tq

ψ

∆gpy,tq

dm ptq

(25)

up t q

where A “ ´µ1 x12 x2 ´ α1 p1 ` η1 cos2ω1 tq x1 ´ β1 x13 ´ λ1 x15 ` f 1 cosω1 t, B “ ´µ1 y21 y2 ´ α1 p1 ` η1 cos2ω1 tq y1 ´ β1 y31 ´ λ1 y51 ` f 1 cosω1 t and ui ptq for i “ 1, 2 are the controllers which govern as per the rule (Equation (21)). Furthermore, during simulation, the initial values of states vectors in master and slave systems are chosen as: x1 p0q “ 0.1, x2 p0q “ 0.2, y1 p0q “ 2.2, y2 p0q “ 0.05, respectively; sliding surface parameters: λ1 “ 25, λ2 “ 10 and switching gain constants: k1 “ 1, k2 “ 5. Therefore, using Equation (19), the error dynamics can be expressed as: .

e1 “ e2 ` 0.5 psinx1 ` siny1 q ` 0.2sint ´ u1 ptq, . e2 “ µ1 e1 ´ µ1 px12 x2 ` y21 y2 q ´ α1 t1 ` η1 cosp2ω1 tqu e1 ´ β1 px13 ´ y31 q ´λ1 px15 ´ y51 q ` 0.5 psinx2 ` siny2 q ` 0.2sint ´ u2 ptq.

(26)

` ˘ ˆ i ` αˆ i ` βˆ i signpsi q ` k i tanhpεsi q for i “ 1, 2 and the where ui ptq “ f i pxq ´ gi pyq ` Fi pxqθˆ i ´ Gi pyqψ unknown parameters have been taken as per Equation (22). The following are the time series of synchronization errors (Figure 14), update parameters (Figures 15 and 16), states vectors (Figure 17) and for secure communication scheme (Figures 18 and 19).

 1 ( x1  y1 )  0.5  sin x2  sin y2   0.2sin t  u2 (t ).





ˆ  G ( y ) ˆ i  ˆ i  ˆ i sign( si )  ki tanh(si ) where ui (t )  fi ( x)  gi ( y)  Fi ( x) i i

for i  1, 2

and the unknown parameters have been taken as per Equation (22). The following are the time series 12 of 21 errors (Figure 14), update parameters (Figures 15 and 16), states vectors (Figure 17) and for secure communication scheme (Figures 18 and 19).

Systems 2016, 4, 25 of synchronization

0.0 0.5 1.0 1.5 e1 e2

2.0 0.0

Systems 2016, 4, 25 Systems 2016, 4, 25

30 30

0.2 0.4 0.6 0.8 Fig 14: Time Series of e1 & e2

1.0

13 of 22 13 of 22

Figure 14. Time Series of e1 & e2 . Figure 14. Time Series of e1 & e2 .

1 1 2

20 20

2 1 1 2

10 10

2

0 0 10 0 10 0

5 10 15 5 Time Series 10of , , 15 Fig 15: 1 2 1& 2 ˆ &ˆ . Figure Fig 15. 15:Time Time Series Seriesof of ˆˆ11,,ˆˆ22,, Figure 15. Time Series of θ1ˆ, θ2ˆ, ψˆ 11 & ψˆ 22 . Figure 15. Time Series of 1 , 2 ,ˆ1 &ˆ 2 .

20 20

30 30

1 21

20 20

2 1 1 2

10 10 0 00 0

3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.00.0 0.0

2

5 10 15 5 10 Fig 16: Time Series of 1 , 2 , ˆ 1 15 & ˆ2 Figure 16. 16. Time Series Series of αˆ1ˆ , ,αˆ2ˆ, β ˆ2 . 1ˆ & β Figure Fig 16:Time Time Seriesof of 11, 22,, 11 & &  22 . ˆ ˆ ˆ Figure 16. Time Series of 1 ,  2 , 1 & ˆ2 .

20 20

x1 yx11 xy21 yx22 y2

0.1 0.2 0.3 0.4 0.1 17: Time0.2 Fig Series of x0.3 y2 1 , y1 , x2 &0.4 Figure 17. Time Time Series Series of of x1x,1 ,yy11, ,xx22 & & yy22 . Fig 17: Figure 17. Time Series of x1 , y1 , x2 & y2 .

0.5 0.5

0 0

5

10

15

20

Fig 16: Time Series of 1 , 2 , 1 & 2 Figure 16. Time Series of ˆ1 , ˆ 2 , ˆ1 & ˆ2 .

Systems 2016, 4, 25

3.0

13 of 21

x1

2.5

y1

2.0

x2 y2

1.5 1.0 0.5 0.0 0.0

Systems 2016, 4, 25 Systems 2016, 4, 25

0.1 0.2 0.3 0.4 Fig 17: Time Series of x1 , y1 , x2 & y2 Figure Figure17. 17.Time TimeSeries Seriesof ofx1x,1y,1y,1x, 2x2&&yy22. .

0.5

14 of 22 14 of 22

3 3 2 2 1 1 0 0 1 1 2

x1

2 3 3

xs1 s

0 0

2 4 6 8 Series6of x1 & s 8 2 Fig 18: Time 4 Figure & ss . Fig 18. 18: Time Time Series Seriesof of xx11 & Figure 18. 18. Time Time Series Figure Series of of xx11 & & ss..

10 10

0.0 0.0 0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0

m m m'

0 0

1 2 3 4 1 Fig 19: Time 2 Series3of m & m' 4 Figure 19. Time Time Series of m & m1'. . Figure Fig19. 19: TimeSeries Seriesofof m m& &m m' Figure 19. Time Series of m & m ' .

m'

5 5

6 3.3. 3.3. Synchronization Synchronization of of Two TwoIdentical IdenticalΦ Φ6—DO —DO Using Using RASMC RASMC 6 6 3.3. Synchronization of Two Identical Φ —DO Using RASMC In In this this section, section, we we synchronize synchronize the the identical identical pairs pairsof ofchaotic chaoticΦΦ6—DO —DO using using RASMC RASMC under under the the effect of external uncertainty and external disturbance for both master and slave systems. After adding 6 section,uncertainty we synchronize the identical pairs offor chaotic —DO and usingslave RASMC under the effectInofthis external and external disturbance both Φmaster systems. After the external uncertainty and disturbances, Equation (2)for can be written as as a slave pair and adding external uncertainty andexternal disturbances, Equation (2) can be written a pairof ofmaster masterAfter and effect ofthe external uncertainty and disturbance both master and systems. slave adding the external uncertainty and disturbances, Equation (2) can be written as a pair of master and slave systems: systems: « ff « ff« ff « ff « ff slave systems: x1 1 0.1sin t . 0  0  x2x2 0 0   11   0.5sin 0.5sinx 0.1sint x“ x `  ` (27)  µ12 `0.5sin xx1 2  0.1sin tt  A  0  0x02 ´x0 0.5sinx 0.1sint 2x  (27) loomoon loooooomoooooon looooooomooooooon  2 looomooon 2    0.5sin 2    0.1sin  x   Alooooooomooooooon       0.1sinmt  (27) ( x) fpxq f A  m d ptq f (∆f x ,ptx,t )x2q   0FpFx,t( qx,t )x2  θ2  0.5sin d (t )

f ( x)

F ( x ,t )



f ( x ,t )

d m (t )

0   1   0.5sin y   0.1sin t   u (t )  0 y y   0    y 2 0   1    0.5sin y1   0.1sin t   u1(t )  y   B   02  y2   2    0.5sin y12    0.1sin t   u12 (t ) t  u2u ((tt) )  gB( y)  0 G ( y,t )y2  2   0.5sin  0.1sin g ( y ,t ) y2  d m (t )

(28) (28)

Systems 2016, 4, 25

14 of 21

«

ff « ff« ff « ff « ff « ff 0 y2 0 1 ´0.5siny1 ´0.1sint u1 ptq y“ ` ` ` ` B 0 ´y2 ´0.5siny2 µ2 ´0.1sint u2 ptq loomoon looooooomooooooon loooooooomoooooooon looomooon looooooomooooooon loooomoooon .

gpyq

Gpy,tq

dm ptq

∆gpy,tq

ψ

(28)

uptq

where Systems 2016, 4, 25 Systems 2016, 4, 25

A “ ´α2 t1 ` η2 cosp2ω2 tqu x1 ´ β2 x13 ´ λ2 x15 ` f 2 cosω2 t, B“

´α2 t1 ` η2 cosp2ω2 tqu y1 ´ β2 y31

´ λ2 y51

` f 2 cosω2 t

15 of 22 15 of 22

x1 (0)  0 , x2 (0)  1.5 , y1 (0)  0.5 , y2 (0)  1 , respectively; sliding surface parameters: and ui ptq for i “ 1, 2 are the controllers which govern as per the rule (Equation (22)). Furthermore, and , y1 (0) gain , y2 (0)  sliding surface parameters:   12 k11, krespectively; x11(0)  0, , 2 x210 (0);  1.5switching  0.5constants: 2  20 .

during simulation, the initial values of states vectors in master and slave systems are chosen as: using Equation (19), theconstants: error dynamics as: , x 2p0q ; and .surface parameters: Therefore, 12  10 k1  kcan be 20expressed 2sliding x11p0q “ 0, “ 1.5, y1switching p0q “ 0.5, ygain λ1 “ 12, λ2 “ 10; 2 2 p0q “ 1, respectively; Therefore,gain using error can be expressed as: and switching “ kxthe “sin 20. y dynamics 2 e1constants: Equation e2  0.5k1(19), sin 1 1   0.2sin t  u1 (t ), Therefore, using Equation (19), the error dynamics can be expressed as: ee1  e 0.5sin1x1sin y1  0.2sin t  u1 (3t ), y 3 )   ( x 5  y 5 ) 2 2 e (29) 1 2 1 cos(2 1t ) e1  1 ( x 1 1 1 1 1 . 2 3 3 5 5 q ` 0.2sint ´ u ee1 “  e2 ` 0.5 psinx ` siny ptq, 1  cos(2 1 x  y )   (x  y )  2x1 (29) . 2  0.5 sin sin1 y2 1   0.2sin 2 e1  1t ) e1 tu 1( 13 1 15 15 3 2 (t1). (29) e2 “ ´µ2 e1 ´ α22t1 ` η1 cosp2ω 1 tqu e1 ´ β1 px1 ´ y1 q ´ λ1 px1 ´ y1 q `0.5 0.5 psinx sin x22`siny sin 2yq2` 0.2sint 0.2sin´tu2 ptq. u2 (t ).  ˆ  G ( y ) ˆ i  ˆ i  ˆ i sign( si )  ki tanh(si ) for i  1, 2 where ui (t )  fi ( x)  gi ( y)  Fi ( x) i i ` ˘ ˆ Gi pyq ˆ as ˆβˆi i(22). where ´ `) Fi pxq ˆi per ` 1, 2series and for ) f ifpxq ) gigpyq Fi (θˆxi )´ Gtaken (ˆ yi ` ) signps ˆ i The sign( ski i)tanhpεs  ki tanh( the sii)“time i  1,the 2 and theuiuptq unknown parameters have been following are of iq ` i q for i (t “ i (x i(y i iψ iα unknown parameters have been20), taken as per (22). The following are the time series vectors of synchronization synchronization errors (Figure update (Figures 21 and 22), states (Figure and the unknown parameters have been parameters taken as per (22). The following are the time series23) of errors (Figure 20), update parameters (Figures 21 and 22), states vectors (Figure 23) and for secure and for secure communication scheme (Figures 24 25). synchronization errors (Figure 20), update parameters (Figures 21 and 22), states vectors (Figure 23) communication scheme (Figures 24 and 25). and for secure communication scheme (Figures 24 and 25).

 

 

e1

0.4

ee12

0.4 0.2

e2

0.2 0.0 0.0 0.2 0.2 0.4 0.4 0.0 0.0

0.2 0.4 0.6 0.8 of e1 & e20.8 0.2 Fig 20: Time 0.4 Series0.6 Figure 20. Time Series of e1 & e2 . Fig 20: Time Series of e1 & e2 Figure 20. Time Series of e & e2 . Figure 20. Time Series of e11 & e2 .

1.0 1.0

4 4

1

3

12 21

3

12

2

2

2 1 1 0 0

0

0

5

10

15

20

Time of 1ˆ , ˆ2 , 1 & 20 5Fig 21: 10Series Figure 21. Time Series of 15 θ1 , θ2 , ψˆ 1 & ψ2ˆ 2 . ˆ &ˆ . Figure 21.Time TimeSeries Seriesofof 1,ˆ1 ,2,ˆ2 , Fig 21: 1 1& 2 2

Figure 21. Time Series of ˆ1 ,ˆ2 ,ˆ1 &ˆ 2 .

25 25

Systems 2016, 4, 25

Systems 2016, 4, 25 Systems 2016, 4, 25 Systems 2016, 4, 25

16 of 22

16 of 22 15 of 21 16 of 22

4.5 4.04.5 4.5 3.54.0 4.0

1

3.03.5 3.5

2 1

2.53.0 3.0

4

2

2

2 1

2.02.5 2.5 0 2.0 2.0 0 0

1

1

5

10

15

Fig 22: Time Series of 1 , 2 , 1 & 2 5 Time Series 10of ˆ , ˆ , ˆ15& ˆ . Figure 22. 1 2 151 2 5 10 Fig 22: Time Series of 1 , 2 , 1 & 2 ˆ ˆ Fig 22:22. Time Series of of1 ,ˆ12,,ˆˆ 21,  &1 & Figure Time Series ˆ 2 2 . Figure 22. 22. Time Time Series Series of 2ˆ, β 1ˆ & βˆ2 . Figure of αˆ1ˆ,1 ,αˆ 2 , 1 &  2 .

1 2

20 20

4 4

2 2

2

0 0

0

x1 y1x1

2

x1

2 2

x2y1

y1

y2x2

4

x2

0.0 4 4 0.0 0.0

y2

y2

0.2 0.4 0.6 0.8 1.0 0.223: Time0.4 0.8 1.0 Fig Series of x0.6 , y , x & y 1 1 2 2 0.2 0.4 Series Fig 23: of0.6 x1 , xy11,, yx12,0.8 y2 y2 . 1.0 x&2 & Figure 23.Time Time Series of Figure Time Series x, 1y,1yx, 1x,, 2xy2& &y y2 . Fig 23:23. Time Series of of x1of Figure 23. Time Series 1 1 , x2 2& y2 . Figure 23. Time Series of x1 , y1 , x2 & y2 .

3 3 32 21 10 01 12

2 1 0 1 2

x

x11 x1 s s s

2

33 3 00 0

11 22 33 44 1 Fig 3 x11 & ss4 Fig24: 24:2Time Time Seriesof Seriesof Figure 24. Time Series ofxof s. ss .. Figure 24. Time Series & Fig 24: Time Seriesof Figure 24. Time Series xs1 & 1x1&& Figure 24. Time Series of x1 & s .

20

2

55 5

Systems Systems 2016, 2016, 4, 4, 25 25

16 of of 22 21 17

0.0 0.1 0.2 0.3 0.4

m m'

0.5 0

5

4 3 2 Fig 25: Time Series of m & m' Figure 25. Time Series of m & m1 . Figure 25. Time Series of m & m ' . 1

3.4. Synchronization Synchronization of of Φ Φ66—VDPO —VDPOand andDO DOUsing UsingRASMC RASMC 3.4. 6 —VDPO and DO using In this In this section, section, we we synchronize synchronize the the non-identical non-identical pairs pairs of of chaotic chaotic Φ Φ6—VDPO and DO using RASMC under under the the effect effect of for both RASMC of external external uncertainty uncertainty and and external external disturbance disturbance for both master master and and slave slave systems. After adding the external uncertainty and disturbances, Equations (1) and (2) can be written systems. After adding the external uncertainty and disturbances, Equations (1) and (2) can be written as aa pair pair of as of master master and and slave slave systems, systems, respectively: respectively: « ff « ff« ff « ff « ff x1 1 0.1sin t . 0  0  xx 0.5sinx 0.1sint 2 2 0 0   11   0.5sin x“ x `  ` (30) 2  `  A  A 00 x2x  µ1  0.5sin 0.5sinx 0.1sint (30) x 0.1sin t loomoon looomooon looooooomooooooon loooooomoooooon   loooooomoooooon   2 1 2      fpxq f ( x )

FpFx,t ( xq ,t )

θ

f (∆f x ,tp)x,tq

m d m ( t )d ptq

ff « ff« ff « ff « ff « ff 0.5sin y1   0.1sin t   u1 (t )u1 ptq 0  0  y2 y2 0 0  11   ´0.5siny ´0.1sint y “ y  `    ` ` ´0.1sint  ` u ptq  µ2   B 0 0 ´y ´0.5siny 2 y 0.5sin y22    (t ) 2 0.1sin t  u2loooomoooon loomoon B looooooomooooooon looooooomooooooon 2looomooon    2  loooooooomoooooooon «

.

gpyq g ( y )

where where

G (q y ,t ) Gpy,t

ψ

m

∆g g ( pyy,t ,t ) q

d m d( t ) ptq

(31) (31)

u ( t ) uptq

A “ ´µ x2 x ´ α t1 ` η cosp2ω tqu x ´ β x3 ´ λ x5 ` f cosω t

A  11x121x22  111  11cos(211t ) x11 11x131 11x151 f11cos 11t B “ ´α2 t1 ` η2 cosp2ω2 tqu y1 ´ β2 y31 ´ λ2 y51 ` f 2 cosω2 t

B  2 1  2 cos(22t ) y1 2 y13  2 y15  f 2 cos 2t

and ui ptq for i “ 1, 2 are the controllers which govern as per the rule (Equation (22)). Furthermore, during initial of states slave systems chosen as: and the values controllers whichvectors governin as master per the and rule (Equation (22)).are Furthermore, 2 are ui (simulation, t ) for i  1,the x1 p0q “ 0.1, x2 p0q “ 0.2, y1 p0q “ 0, and y2 p0q “ 1.5, respectively; sliding surface parameters: λ1 “ 12, during simulation, the initial values of states vectors in master and slave systems are chosen as: λ2 “ 10; and switching gain constants: k1 “ k2 “ 20. , respectively; sliding surface parameters: x1 (0)Therefore,  0.1 , x2using (0) Equation 0.2 , y1 (0)  0 , and y2 (0)  1.5 (19), the error dynamics can be expressed as: 1  12 ,  2  10 ; and switching gain constants: k1  k2  20 . . eTherefore, psinx q ` 0.2sint ´ udynamics 1 “ e2 ` 0.5 1 ` siny1(19), 1 ptq, using Equation the error can be expressed as: . 2 (32) e2 “ µ1 p1 ´ x1 qx2 ´ α1 t1 ` η1 cosp2ω1 tqu x1 ´ β1 x13 ` β2 y31 ´ λ1 x15 ` λ2 y51 ` f 1 cosω1 t 0.2sin (t ), t ` 0.5 psinx ` siny q ` 0.2sint ´ u ptq.  sin x  sin y1  2tqu `µe21y2 e`2 α0.5 y1 `t fu2 1cosω 2 t1 ` η11 cosp2ω 2 2 2 2 e2  1 (1  x12 ) x2  1 1  1 cos(21t ) x1  1 x13  2 y13  1 x15   2 y15  f1 cos 1t (32) ` ˘ ˆ ˆ ˆ where ui ptq “ f i pxq ´ g pyq ` F pxq θ ´ G pyq ψ ` α ˆ ` β signps q ` k tanhpεs q for i “ 1, 2 and the i 2t )i  y1  i f 2 cos  i 2t  0.5 i t  u2 (t ). i  sin x2 i sin yi2   0.2sin   2 y2 i  2 1  i1 cos(2 unknown parameters have been taken as per Equation (22). The following are the time series of ˆ ˆ 27 and 28), states vectors (Figure 29) synchronization (Figure 26), update parameters ˆ ˆ(Figures where ui (t )  ferrors i ( x)  gi ( y )  Fi ( x)i  Gi ( y )i  i  i sign( si )  ki tanh(si ) for i  1, 2 and for secure communication scheme (Figures 30 and 31) as well as the derivative of Lyapunov . and the unknown parameters have been taken as per Equation (22). The following are the time series function pVptqq for all three pairs together (Figure 32). of synchronization errors (Figure 26), update parameters (Figures 27 and 28), states vectors (Figure 29) and for secure communication scheme (Figures 30 and 31) as well as the derivative of





Lyapunov function (V (t )) for all three pairs together (Figure 32).

Systems 2016, 4, 25

18 of 22

Systems 2016, 4, 25 Systems Systems 2016, 2016, 4, 4, 25 25

18 of 22 17 18 of of 21 22

0.0 0.0 0.2 0.0 0.2 0.4 0.2 0.4 0.6 0.4 0.6 0.8 0.6 0.8 1.0 0.8 1.0 1.2 1.0 1.2 1.20.0 0.0 0.0

6 6 5 6 5 4 5 4 3 4 3 23 2 1 2 1 0 1 0 1 0 1 0.0 1 0.0 0.0

e1

0.1 0.2 0.3 0.4 of e1 & e2 0.4 0.1Fig 26: Time 0.2 Series 0.3 0.1 Fig 0.2 Series 0.3 e2e.20.4 Figure 26.26: Time of Time Series ofe1e& 1& Fig 26: Time Series of e & Figure 26. Time Series of e1 & ee2 . Figure 26. Time Series of e 1& e 2. Figure 26. Time Series of e11 & e22 .

0.5 0.5

1 21 1 12 2 21 1 2 2

0.2 0.4 0.6 0.8 0.2 27: Time0.4 Fig Series of 10.6 , , & 0.8 2 ˆ 2ˆ 1ˆ  0.2 27. Time0.4 0.6 0.8 ˆ Figure Fig 27: Time Series Seriesof of ˆ11,,ˆ22,,ˆ 11 & & ˆ22 . Figure 27. Time Series of & ψˆ2 . . 1,ˆ, θ 2ˆ,,2ψ,11ˆ1& & Figure 27.Time TimeSeries Seriesof of θ1 Fig 27: 1 ,2 22 Figure 27. Time Series of ˆ1 ,ˆ2 ,ˆ1 &ˆ 2 .

9 9 89 8 78 7 67 6 56 5 45 4 34 0.0 3 30.0 0.0

e2e1 e1 e2 e0.5 2

1.0 1.0 1.0

1 21 1 12 2 21 1 2 2

0.2 0.4 0.6 0.8 0.2 0.4 0.6 Fig 28: Time Series of 1 , 2 , ˆ1 & 0.8 ˆ 2 , β 1 & 0.8 Figure βˆ2 2 . 1, α 0.2 28. Time 0.4Series of αˆ0.6 ˆ Fig 28: Time Series of , , & Figure 28. Time Series of ˆ11 , ˆ22 , 11 & ˆ22. Fig 28: Time Series of 1 , 2 , 1ˆ & ˆ2 Figure 28. Time Series of ˆ1 , ˆ 2 ,  1 & 2 . Figure 28. Time Series of ˆ1 , ˆ 2 , ˆ1 & ˆ2 .

1.0 1.0 1.0

Systems 2016, 4, 25

19 of 22

3 Systems2016, 2016,4,4,25 25 Systems Systems 2016, 4, 25

19of of21 22 18 19 of 22

2 3 3

1

x1

2 2

y1 x2 x1y x1 2 y1 y1 0.2 0.4 0.6 x2 0.8 x Fig 29: Time Series of x1 , y1 , xy22 & y2 Figure 29. Time Series of x1 , y1 , yx222 & y2

0 1 1

0.0 0 0

0.0 0.0

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Fig 29: Time Series of x1 , y1 , x2 & y2 Fig 29: Figure 29.Time TimeSeries Seriesofofx1x, 1y,1y,1x, 2x2&&yy2 2 . Figure 29. Time Series of of x1x1, ,yy11, ,xx22 & & y22 ..

3

1.0

. 1.0 1.0

2 3

1 3 0 1

2 2 1 1

0 0 2 1 1

x1 s

3 2 2 03 3 0 0

x1 x1 s s

2

4 6 8 Fig 30: Time Series6of x1 & s 8 2 4 2 4 Figure 30. Time Series6 of x1 & s8. Fig 30: Time Series of x1 & s Fig 30: Time Time Series of of x & ss. Figure Figure 30. 30. Time Series Series of x1x1 & &s . Figure 30. Time Series of x11 & s .

0.10

10 10 10

m

0.10 0.10

m m m' m' m'

0.05 0.05 0.05

0.00

0.00 0.00

0.05 0.05 0.05

0 0 0

33 44 11 22 1 2 3 4 Fig 31: Time Series of m & m' Fig 31: Time Seriesofofmm&&m' m' 1. Fig31.31: Time Series Figure Time mm&&m mm ' .' . Figure31. 31.Time Time Series Series of of m & Figure Series of Figure 31. Time Series of m & m ' .

5 5 5

Systems 2016, 4, 25

19 of 22 21 20

0 500 1000 1500 2000 2500

V t for 6 VP & DO

3000

V t for 6 VP V t for 6 DO

3500 0.0

0.1

0.2

0.3

0.4

0.5

Series of.

Fig 32: Time V t in all cases Figure 32. Time Series of Vptq in all cases. Figure 32. Time Series of V (t ) in all cases.

3.5. Numerical Simulations and Discussion 3.5. Numerical Simulations and Discussion The main aim of Section 3 was to implement the RASMC on the two identical pairs of Φ6 —VDPO The main aim of Section 3 was to implement the RASMC on the two identical pairs of and DO and one non-identical pair of Φ6 —VDPO and DO for synchronization purpose. It has been Φ6—VDPO and DO and one non-identical pair of Φ6—VDPO and DO for synchronization purpose. observed that not only RASMC is found to be very effective for all the three pairs under consideration; It has been observed that not only RASMC is found to be very effective for all the three pairs under it is also effective for a secure communication scheme. To the best of our knowledge, this has been consideration; it is also effective for a secure communication scheme. To the best of our knowledge, done for the first time using RASMC. The time of stabilization of synchronization is very short (nearly this has been done for the first time using RASMC. The time of stabilization of synchronization is t “ 0.2) for all the cases that can be observed in the plotted time series (Figures 14–32). We can say very short (nearly t  0.2 ) for all the cases that can be observed in the plotted time series now that the time response of stabilization of synchronization is much quicker in RASMC technique (Figures 14–32). We can say now that the time response of stabilization of synchronization is much than it was in LAC. quicker in RASMC technique than it was in LAC. 4. Conclusions 4. Conclusions In this computational cum comparative study, the problem of chaotic synchronization of chaotic In computational cum comparative study, the problem ofbeen chaotic synchronization chaotic systemsthis is repeated using Mathematica via LAC technique. It has found that the timeof response systems is repeated using Mathematica via LAC technique. It has been found that the time response of stabilization of synchronization is reduced by half when it is done using Mathematica. On the of stabilization of synchronization is reduced by halfvia when it is done usingresponse Mathematica. On the other other hand, when the same pairs are synchronized RAMSC, the time of stabilization of hand, when the same pairs are synchronized via RAMSC, the time response of stabilization of synchronization is found to be much faster than the LAC technique. Finally, we conclude the following synchronization is found to be much faster than the LAC technique. Finally, we conclude the remarkable features of our proposed study: following remarkable features of our proposed study: (1) The time response of stabilization of synchronization for LAC in our study was found to occur (1) The time response of stabilization of synchronization for LAC in our study was found to occur with rapid convergence if simulation is done with Mathematica. with rapid convergence if simulation is done with Mathematica. (2) For the same pairs of master and slave systems considered in our study, the RASMC is found to (2) For the same pairs of master and slave systems considered in our study, the RASMC is found to be more effective in terms of time response of stabilization of synchronization. be more effective in terms of time response of stabilization of synchronization. On the selection of of appropriate mathematical tools for On the basis basis of of these thesetwo twopoints, points,we weconclude concludethat that selection appropriate mathematical tools simulation and technique for synchronization is very important. for simulation and technique for synchronization is very important. authors would like to to express express their their gratitude gratitude to to the the honorable honorable reviewers reviewers who who suggested suggested Acknowledgments: The authors many worthwhile worthwhile changes changes to to improve improve the many the work work of of this this manuscript. manuscript. Author Contributions: thethe literature review andand proposed the problem. Adyda Binti Author Contributions: Azizan AzizanBin BinSaaban Saabanperformed performed literature review proposed the problem. Adyda Ibrahin and Israr Ahmad performed the analytical analysis and wrote the paper. Mohammad Shahzad performed Binti Ibrahin and Israr Ahmad performed the analytical analysis and wrote the paper. Mohammad Shahzad the numerical simulations and revised the manuscript. performed the numerical simulations and revised the manuscript. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest.

References References 1. 1. 2. 2.

Shahzad, M.; Pham, V.T.; Ahmad, M.A.; Jafari, S.; Hadaeghi, F. Synchronization and circuit design of a chaotic Shahzad, M.;coexisting Pham, V.T.; Ahmad, M.A.; Eur. Jafari, S.; Hadaeghi, Synchronization and[CrossRef] circuit design of a system with hidden attractors. Phys. J. Spec. Top.F.2015, 224, 1637–1652. chaotic system with coexisting hidden attractors. Eur. Phys. J. Spec. Top. 2015, 224, 1637–1652. Yao, L.; Cai, G. Chaos Synchronization of a New Hyperchaotic Finance System Via a Novel Chatter Free Yao, L.; Mode Cai, G.Control Chaos Strategy. Synchronization of a New Finance System Via a Novel Chatter Free Sliding Int. J. Nonlinear Sci.Hyperchaotic 2014, 17, 176–181. Sliding Mode Control Strategy. Int. J. Nonlinear Sci. 2014, 17, 176–181.

Systems 2016, 4, 25

3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23.

24.

25. 26.

20 of 21

Mobayen, S. A Novel Global Sliding Mode Control Based on Exponential Reaching Law for a Class of under Actuated Systems with External Disturbances. J. Comput. Nonlinear Dyn. 2015, 11, 021011–021019. [CrossRef] Azar, A.T.; Zhu, Q. Advances and Applications in Sliding Mode Control Systems; Springer International Publishing: Cham, Switzerland, 2014. Shahzad, M.; Raziuddin, M. Synchronization Of Three Dimensional Cancer Model With Rosslar System Using A Robust Adaptive Sliding Mode Controller. Int. J. Math. Arch. 2015, 6, 123–132. Shahzad, M.; Saaban, A.B.; Ibrahim, A.B.; Ahmad, I. Adaptive control to synchronise and anti-synchronise two identical time delay Bhalekar-Gejji chaotic systems with unknown parameters. Int. J. Autom. Control 2015, 9, 211–227. [CrossRef] Ma, J.; Zhang, A.; Xia, Y.; Zhang, L. Optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems. Appl. Math. Comput. 2010, 215, 3318–3326. [CrossRef] Lin, C.; Peng, Y.; Lin, M. CMAC-based adaptive backstepping synchronization of uncertain chaotic systems. Chaos Soliton Fract. 2009, 42, 981–988. [CrossRef] Bai, E.W.; Lonngren, K.E. Synchronization of two Lorenz system using Active control. Phys. Rev. Lett. 1997, 64, 1199–1196. [CrossRef] Vincent, U.E. Synchronization of Identical and Nonidentical 4-D Chaotic Systems using Active Control. Chaos Solitons Fract. 2008, 37, 1065–1075. [CrossRef] Njah, A.N. Synchronization and Anti-synchronization of double hump Duffing-Van der Pol Oscillators via Active Control. J. Inf. Comput. Sci. 2009, 4, 243–250. Njah, A.N. Synchronization via active control of parametrically and externally excited Φ6 —Van der Pol and duffing oscillators and application to secure communications. J. Vib. Cont. 2011, 17, 493–504. [CrossRef] Shahzad, M.; Ahmad, I. Experimental study of synchronization & Anti-synchronization for spin orbit problem of Enceladus. Int. J. of Cont. Sci. Eng. 2013, 3, 41–47. Pisarchik, A.N.; Arecchi, F.T.; Meucci, R.; Garbo, A.D. Synchronization of Shilnikov Chaos in CO2 Laser with Feedback. Laser Phys. 2014, 11, 1235–1239. Hammami, S. State feedback-based secure image cryptosystem hyperchaotic synchronization. ISA Trans. 2015. [CrossRef] [PubMed] Rafikov, M.; Balthazar, J.M. On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun. Nonlinear Sci. Numer. Simul. 2008, 13, 1246–1255. [CrossRef] Luo, R.Z.; Zhang, F. Finite-Time Modified Projective Synchronization of Unknown Rossler and Coullet Systems. Commun. Control Sci. Eng. 2013, 1, 51–56. Cai, N.; Jing, Y.; Zhang, S. Modified projective synchronization of chaotic systems with disturbances via active sliding mode control. Commun. Nonlinear Sci. Numer. Simul. 2010, 15, 1613–1620. [CrossRef] Fu, G. Robust adaptive modified function projective synchronization of different hyperchaotic systems subject to external disturbance. Commun. Nonlinear Sci. Numer. Simul. 2012, 17, 2602–2608. [CrossRef] Ahmad, I.; Saaban, A.; Ibrahim, A.; Shahzad, M. A Research on the Synchronization of Two Novel Chaotic Systems Based on a Nonlinear Active Control Algorithm. Eng. Tech. Appl. Sci. Res. 2015, 5, 739–747. Singh, P.P.; Singh, J.P.; Roy, B.K. Synchronization and anti-synchronization of Lu and bhalekar-Gejji chaotic systems using nonlinear active control. Chaos Solitons Fract. 2014, 69, 31–39. [CrossRef] Ahmad, I.; Saaban, A.B.; Ibrahim, A.B.; Shahzad, M. On Globally Exponential Stable Complete Synchronization of Nearly Identical Hyperchaotic Systems via Linear Active Control. Ciênc. Téc. Vitiviníc. Sci. Technol. J. 2015, 30, 141–154. Shahzad, M. The Improved Results with Mathematica and Effects of External Uncertainty & Disturbances on Synchronization using a Robust Adaptive Sliding Mode Controller: A Comparative Study. Nonlinear Dyn. 2015, 79, 2037–2054. Pourmahmood, M.; Khanmohammadi, S.; Alizadeh, G. Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller. Commun. Nonlinear Sci. Numer. Simul. 2011, 16, 2853–2868. [CrossRef] Vaidyanathan, S.; Rasappan, S. Global Chaos Synchronization for WINDMI and Coullet Chaotic Systems Using Active Control. J. Cont. Eng. Tech. 2013, 3, 69–75. Ahmad, I.; Saaban, A.B.; Ibrahim, A.B.; Shahzad, M. A Research on Active Control to Synchronize a New 3D Chaotic System. Systems 2016, 4. [CrossRef]

Systems 2016, 4, 25

27.

28. 29. 30. 31. 32. 33. 34.

35.

36.

21 of 21

Ahmad, I.; Saaban, A.B.; Ibrahim, A.B.; Shahzad, M.; Naveed, N. The Synchronization of Chaotic Systems with Different Dimensions by a Robust Generalized Active Control. Int. J. Light Electron. Opt. 2016, 127, 4859–4871. [CrossRef] Ahmad, I.; Saaban, A.; Ibrahim, A.; Shahzad, M. Global Chaos Identical and Nonidentical Synchronization of a New Chaotic System Using Linear Active Control. Complexity 2014. [CrossRef] Siewe, S.M.; Moukam, K.F.M.; Tchawoua, C.; Woafo, P. Bifurcation and chaos in triple-well Φ6 —Van der Pol oscillator driven by external and parametric excitations. Phys. A 2005, 357, 383–396. [CrossRef] Tchoukuegno, R.; Nbendjo, B.R.N.; Woafo, P. Resonant oscillations and fractal basin boundaries of a particle in a Φ6 potential. Phys. A 2002, 304, 362–368. [CrossRef] Tchoukuegno, R.; Nbendjo, B.R.N.; Woafo, P. Linear feedback and parametric controls of vibrations and chaotic escape in a Φ6 potential. Int. J. Nonlinear Mech. 2003, 38, 531–541. [CrossRef] Li, W.; Chang, K. Robust synchronization of drive-response chaotic systems via adaptive sliding mode control. Chaos Soliton Fract. 2009, 39, 2086–2092. [CrossRef] Yan, J.; Hung, M.; Chiang, T.; Yang, Y. Robust synchronization of chaotic systems via adaptive sliding mode control. Phys. Lett. A 2006, 356, 220–225. [CrossRef] Aghababa, M.P.; Heydari, A. Chaos synchronization between two different chaotic systems with uncertainties, external disturbances, unknown parameters and input nonlinearities. Appl. Math. Model. 2012, 36, 1639–1652. [CrossRef] Aghababa, M.P.; Akbari, M.E. A chattering-free robust adaptive sliding mode controller for synchronization of two different chaotic systems with unknown uncertainties and external disturbances. Appl. Math. Comput. 2012, 218, 5757–5768. [CrossRef] Khan, A.; Shahzad, M. Synchronization of circular restricted three body problem with Lorenz hyper chaotic system using a robust adaptive sliding mode controller. Complexity 2013, 18, 58–64. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).