Control of a new chaotic fractional-order system using ...

NONLINEAR STUDIES - www.nonlinearstudies.com Vol. 22, No. 4, pp. 1-13, 2015 c CSP - Cambridge, UK; I&S - Florida, USA, 2015 ⃝

Control of a new chaotic fractional-order system using Mittag-Leffler stability ⋆

Z.Hammouch 1 , T.Mekkaoui 1 1

E3MI, D´epartement de Math´ematiques, FST Errachidia Universit´e Moulay Ismail BP.509 Boutalamine 52000 Errachidia, Morocco ⋆

Corresponding Author. E-mail address: [email protected]

Abstract. In this paper an active control method is used to suppress chaos to unstable equilibria based on the Mittag-Leffler stability theory. Using a Multistep Generalized Differential Transform Method, numerical simulations are shown to verify the analytical results.

1 Introduction Fractional calculus is a powerful mathematical tool, its application to engineering and modeling of physical systems has attracted much attention in recent years. This theory generalizes the classical differentiation and integration into noninteger order ones [23]. It has been found that in interdisciplinary fields, many systems can be described more accurately and more conveniently by fractional differential equations (FDEs) see, for example, [20] and the references therein. On the other hand, Chaos theory is a field of study in mathematics, with applications in several disciplines including meteorology, sociology, physics, engineering, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions. Based on the extension of applications of FODEs, fractional-order chaotic systems have stimulated intense attentions in recent years due to their applications in secure communication, encryption and control processing, DNA computiong, etc. However, chaos is undesirable in many applications, therefore it is necessary to control the chaotic behaviours of dynamical systems. Recently, study on chaos control of fractional-order chaotic systems has become an active research field. Several techniques have been devised for chaos control, but most are developments of two basic approaches: the OGY (Ott, Grebogi and Yorke) method [19], and Pyragas continuous control [22], both methods require a previous determination of the unstable periodic orbits of the chaotic system before the controlling algorithm can be designed. In addition, another methods have been developed to stabilize chaotic systems such as adaptive control [3] and feedback control [10], etc. In this work we investigate the problem of chaos control for a new chaotic dynamical system with fractional derivatives, and propose a simple active control method using Mittag-Leffler stability theory. The present control technique is simple and easy to use. This paper is organized as follows : section 2 presents some preliminary results, section 3 gives the 2010

Mathematics Subject Classification: Primary: 34H10, 26A33. Secondary: 65P20. Keywords: Fractional chaotic system, Active control, MSGDTM, Mittag-Leffler stability.

2

Z.Hammouch, T.Mekkaoui

dynamical behaviour and the numerical simulation of the fractional autonomous system. Section 4 presents chaos control. Finally, in section 5, concluding comments are given.

2 Preliminaries 2.1 Fractional calculus Fractional calculus is a generalization of integration and differentiation. Its idea has been known since the development of the regular calculus, with the first reference probably being associated with Leibniz and L’Hˆopital in 1695 where half-order derivative was mentioned. Historical introductions on FDEs, can be found in [18]-[21]. Commonly used definitions for fractional derivatives are due to Riemann-Liouville, and Caputo [2]. In what follows, Caputo derivatives are considered, taking the advantage that this allows for traditional initial and boundary conditions to be included in the formulation of the considered problem. Definition 2.1. A real function f (x), x > 0, is said to be in the space Cµ , µ ∈ R if there exits a real number λ > µ such that f (x) = xλ g(x), where g(x) ∈ C[0, ∞) and it is said to be in the space Cµm if and only if f (m) ∈ Cµ for m ∈ IN . Definition 2.2. The Riemann-Liouville fractional integral operator of order α of a real function f (x) ∈ Cµ , µ ≥ −1, is defined as ∫ x 1 α (x − t)α−1 f (t)dt, α > 0, x > 0 and J 0 f (x) = f (x). (2.1) J f (x) = Γ (α) 0 The operators J α has some properties, for α, β ≥ 0 and ξ ≥ −1 : • J α J β f (x) = J α+β f (x), • J α J β f (x) = J β J α f (x), (ξ+1) • J α xξ = Γ Γ(α+ξ+1) xα+ξ . Definition 2.3. The Caputo fractional derivative Dα of a function f (x) of any real number α such m in the terms of J α is: that m − 1 < α ≤ m, m ∈ IN , for x > 0 and f ∈ C−1 ∫ x 1 α m−α m D f (x) = J D f (x) = (x − t)m−α−1 f (m) (t)dt (2.2) Γ (m − α) 0 and has the following properties for m − 1 < α ≤ m, m ∈ IN , µ ≥ −1 and f ∈ Cµm : • Dα J α f (x) = f (x), ∑ (k) (0+ ) xk , for x > 0, • J α Dα f (x) = f (x) − m−1 k=0 f k! 2.2 The Multistep Generalised Differential Transform Method (MSGCTM) Several author’s used the MSGDTM, see for examples [5][6][17]. To describe its basic idea, we consider the initial value problem for systems of fractional differential equations  α1 Dt x1 = f1 (t, x1 , x2 , ..., xn ),     Dα2 x2 = f2 (t, x1 , x2 , ..., xn ),    t . (2.3) .      .   αn Dt xn = fn (t, x1 , x2 , ..., xn ).

Control of a fractional-chaotic system by Mittag-Leffler stability

3

subject to the initial conditions xi (t0 ) = ci

for

i = 1, 2, ..., n.

(2.4)

Where Dtαi is the Caputo fractional derivative of order αi (0 < αi ≤ 1), for i = 1, 2, ..., n. Let [t0 , T ] be the interval over which we want to find the solution of the initial value problem (2.3),(2.4). In actual applications of the generalized differential transform method (GDTM) [17],the k th -order approximate solution of the initial value problem (2.3),(2.4) can be expressed by the finite series K ∑ xi (t) = Xi (k)(t − t0 )kαi , t ∈ [t0 , T ], (2.5) i=0

where Xi (k) satisfied the recurrence relation  Γ (kαi + 1)   Fi (k, X1 , X2 , ..., Xn ),  Xi (k + 1) = Γ ((k + 1)αi + 1)    Xi (0) = ci ,

(2.6)

and Fi (k, X1 , X2 , ..., Xn ) is the differential transform of function fi (t, x1 , x2 , ..., xn ), for i = 1, 2, ..., n. The concept of the MSGDTM as it was presented in [17], is as follows. Assume that the interval [t0 , T ] is devided in M subintervals [tm−1 , tm ], for m = 1, 2, ..., M of T − t0 equal step size h = by using the nodes tm = t0 + mh. The basic principle is M • First, we apply the GDT M to the initial value problem (2.3),(2.4) over the sub-interval [t0 , t1 ], we will obtain the first approximate solution xi,1 (t), t ∈ [t0 , t1 ], using the initial condition xi (t0 ) = ci , for i = 1, 2, ..., n. • For m ≥ 2 and at each subinterval [tm−1 , tm ], we use the initial condition xi,m (tm−1 ) = xi,m−1 (tm−1 ), and apply the GDTM to the initial value problem (2.3),(2.4) over the subinterval [tm−1 , tm ]. • We repeat the same process, which generates a sequence of solutions xi,m , m = 1, 2, ..., M , for i = 1, 2, ..., n. • Finally, the MSGDTM assumes the following solution  xi,1 (t), t ∈ [t0 , t1 ],     xi,2 (t), t ∈ [t1 , t2 ],    . (2.7) xi (t) = .      .   xi,M (t), t ∈ [tM −1 , tM ], We stress that the main advantage of this method is that the obtained solution converges for wide time regions. 2.3 Stability results Definition 2.4. The Mittag-Leffler function Eα , is defined as Eα (z) =

∞ ∑ n=0

zn , Γ (αn + 1)

(2.8)

where z is a complex variable and α ̸= 0. The Mittag-Leffler function is a generalization of the exponential function, to which it reduces for α = 1.

4

Z.Hammouch, T.Mekkaoui

Now, we consider the following comensurate fractional-order dynamical system  α  D X(t) = f (X(t)), 

(2.9) x(0) = x0 ,

where X(t) = (x1 , x2 , ..., xn )T ∈ Rn , t > t0 , t ∈ [0, T ], α ∈ (0, 1) and F : T × Rn → Rn is continuous in x. According to Matigon [14], we have the following stability theorem Theorem 2.1. For a given comensurate fractional order system (2.9), the equilibria can be obtained by calculating f (x) = 0. These equilibrium points are locally asymptotically stable if all the ∂f απ eigenvalues λ of the Jacobian matrix J = at the equilibrium points satisfy |arg(λ)| > . ∂x 2 Next, we define the Mittag-Leffler stability of solutions of (2.9) as follows Definition 2.5. The solution X(t) to (2.9) is said to be Mittag-Leffler stable if ||X(t)|| ≤ [m(X(t0 ))Eα (−λ(t − t0 ))]b ,

(2.10)

where λ ≥ 0, b > 0, m(0) = 0, m(X) > 0 and m is locally Lipschitz for X ∈ D, where D = {X ∈ Rn : ||X(t)|| < ρ}. Remark 2.1. We notice that the above definition is a generalization of the classical exponential stability. Theorem 2.2. [13] If there exists a scalar function V (t, X(t)) ∈ C [[0, T ] × D, R+ ] , and positive contants c1 , c2 and c3 such that c1 ||X(t)||2 ≤ V (t, X(t)) ≤ c2 ||X(t)||2 ,

(2.11)

Dα V (t, X(t)) ≤ −c3 ||X(t)||2 ,

(2.12)

and for all (t, X) ∈ [0, T ] × D, t ≥ t0 then the solution of (2.9) is Mittag-Leffler stable.

3 A new fractional-order chaotic system 3.1 Dynamical analysis Recently, Li et al. [12] proposed a new chaotic system based on the construction pattern of Chen and Liu chaotic systems, given as  x˙ = a(y − x),      y˙ = (c − a)x + cy − dxz, (3.1)      z˙ = −bz + ey 2 . Where x, y and y are the state variables, a, b, c, d and e are the positives constant parameters. Remark 3.1. For the dynamical system (3.1), we can obtain for α = 1 that ( ) ( ) ( ) ∂ dx ∂ dy ∂ dz ∇V = + = −a − b + c = −11. ∂x dt ∂y dt ∂z dt

(3.2)

then the dynamical system described by (3.1) is one dissipative system, and an exponential condV = e−11t . traction of the system (3.1) is dt

Control of a fractional-chaotic system by Mittag-Leffler stability

More recently, Li and Tong [11] considered a fractional version (3.1), which reads  α D x = a(y − x),      Dα y = (c − a)x + cy − dxz,      α D z = −bz + ey 2 ,

5

(3.3)

where α ∈ (0, 1]. It was found in [11], that for a = 38, b = 3, c = 30 and d = c = 1, system (3.3) exhibits a chaotic behaviour. It has three equilibria and corresponding eigenvalues are shown as follows.  E1 (0, 0, 0) : λ1 = −33.189, λ2 = 25.189, λ3 = −3,        E2 (8.124, 8.124, 22) : λ1 = −17.9176, λ2 = 3.4588 ± 16.3702i, (3.4)      E (−8.124, −8.124, 22) : λ1 = −17.9176, λ2 = 3.4588 ± 16.3702i,   3 with i2 = −1. All the equilibria are unstable and according to [9] a necessary condition for system (3.3) to remain chaotic is keeping α > 0.8674. 3.2 Numerical simulations by MSGDTM Applying the MSGDTM algorithm to the following system  α D x = a(y − x),        α   D y = (c − a)x + cy − dxz,   Dα z = −bz + ey 2 ,        x(0) = x0 , y(0) = y0 , z(0) = z0 ,

(3.5)

gives  Γ (αk + 1)   Xi (k + 1) = [a(Yi (k) − Xi (k))] ,   Γ (α(k + 1) + 1)       [ ]  ∑k Γ (αk + 1) Yi (k + 1) = (c − a)Xi (k) + cYi (k) − d Xi (l)Zi (k − l) , l=0  Γ (α(k + 1) + 1)      ] [  ∑k  Γ (αk + 1)   Yi (l)Yi (k − l) , −bZi (k) + e  Zi (k + 1) = l=0 Γ (α(k + 1) + 1)

(3.6)

where X(k), Y (k), and Z(k) are the differential transforms of x(t), y(t) and z(t), respectively. The differential transform of the initial conditions X(0) = x0 , Y (0) = y0 and Z(0) = z0 . In view of the differential inverse transform, the differential transform series solution for the system (3.5) can be obtained as

6

Z.Hammouch, T.Mekkaoui

 N  ∑    x(t) = X(n)tαn ,     n=0       N  ∑ Y (n)tαn , y(t) =   n=0        N  ∑    Z(n)tαn .   z(t) =

(3.7)

n=0

According the MSGDTM, the series solution for system (3.5) is given by  K  ∑    X1 (n)tαn , t ∈ [0, t1 ],     n=0       K  ∑    X2 (n)tαn , t ∈ [t1 , t2 ],  x(t) = n=0  .      .     .    K  ∑    XM (n)tαn , t ∈ [tM −1 , tM ], ,  

(3.8)

n=0

 K  ∑    Y1 (n)tαn , t ∈ [0, t1 ],     n=0       K   ∑  Y2 (n)tαn , t ∈ [t1 , t2 ],  y(t) =

(3.9)

n=0

 .      .     .    K  ∑    YM (n)tαn , t ∈ [tM −1 , tM ],   n=0

 K  ∑    Z1 (n)tαn , t ∈ [0, t1 ],    n=0        K  ∑    Z2 (n)tαn , t ∈ [t1 , t2 ],  z(t) =

n=0

 .      .     .    K  ∑    ZM (n)tαn , t ∈ [tM −1 , tM ].  

(3.10)

n=0

Where Xi (n), Yi (n) and Zi (n) for i = 1, 2, ..., M satisfy the following recurrence relations:

Control of a fractional-chaotic system by Mittag-Leffler stability

7

 Γ (αk + 1)   [a(Yi (k) − Xi (k))] ,  Xi (k + 1) =  Γ (α(k + 1) + 1)       [ ]  ∑k Γ (αk + 1) (c − a)Xi (k) + cYi (k) − d Xi (l)Zi (k − l) , (3.11) Yi (k + 1) = l=0  Γ (α(k + 1) + 1)      [ ]  ∑k  Γ (αk + 1)   −bZi (k) + e Yi (l)Yi (k − l) ,  Zi (k + 1) = l=0 Γ (α(k + 1) + 1) and

Finally, we start with

 xi (ti−1 ) = xi−1 (ti−1 ),      yi (ti−1 ) = yi−1 (ti−1 ),      zi (ti−1 ) = zi−1 (ti−1 ).

(3.12)

 X0 (0) = x0 ,      Y0 (0) = y0 ,      Z0 (0) = z0 .

(3.13)

Using the recurrence relation given in (3.11) we obtain the multistep solution given in (3.8), (3.9) and (3.10). The MSGDTM is coded in the computer algebra package Maple 17 (see the Appendix). The time range studied in this work is [0, 3000] with a step size ∆t = 0.01. We take the initial condition as x(0) = 0.5, y(0) = 0.2 and z(0) = 15. Figure 1 represents the time series x(t), y(t) and z(t) for α = 0.9 and k = 4. Figure 2 shows the phases portrait of system (3.5) and Figure 3 depicts the strange chaotic attractor x − y − z for (3.5).

(a)

(b)

(c)

Fig. 1: Time series of system (3.3), (a) for x signals, (b) for y signals and (c) for z signals. From the graphical results in Figures 1-3, it is to conclude that the approximate solutions obtained using MSGDTM are in good agreement with those obtained in [11], for the same system, using the improved Adams-Bashforth algorithm.

8

Z.Hammouch, T.Mekkaoui

(a)

(b)

(c)

Fig. 2: Phase portrait plots for system (3.3 for α = 0.9, (a)) in x − y, (b) in x − z and (c) in y − z.

(d)

Fig. 3: The chaotic attractor of (3.3) for α = 0.9..

4 Active control of the fractional-order chaotic system In this section, we investigate the problem of chaos control of the fractional chaotic system (3.5). In order to control it towards equilibrium points E1 , E2 and E3 , let us assume that the controlled fractional-order autonomous system is given by  α D x = a(y − x) + U1 (t),      Dα y = (c − a)x + ey − dxz + U2 (t), (4.1)      α D z = −bz + ey 2 + U3 (t). where Ui (t) (i = 1, 2, 3) are external active control inputs which will be suitably determined later. To stabilize the chaotic orbits in (3.5) to its equilibrium E1 (resp. E2 or E3 ), we need to add the following active controllers to system (3.5)  U1 = −cy,      U2 = (−c − a)y, (4.2)      U3 = (b − a)z − ey 2 + dxy.

Control of a fractional-chaotic system by Mittag-Leffler stability

and

9

 U1 = −cy,      √ √ √ √ U2 = (−c − a)y + 22 dx − 2 c 66 + a 66 + d 66z + 22 d 66,     √  U3 = (b − a)z − 66 e + 22 b − ey 2 − 2 ey 66 + ydx.

(4.3)

 U1 = −cy,      √ √ √ √ U2 = (−c − a)y + 22 dx + 2 c 66 − a 66 − d 66z − 22 d 66,     √  U3 = (b − a)z − 66 e + 22 b − ey 2 + 2 ey 66 + ydx.

(4.4)

We prove the following result Theorem 4.1. Starting from any initial condition, the equilibrium point E1 (resp. E2 or E3 ) of (4.1) is asymptotically stable when the active controllers Ui , i = 1, 2, 3 are defined as in (4.2) (resp. (4.3) or (4.4)). Proof. As a Lyapunov candidate function associated to system (4.1), we consider the quadratic function defined by 1 (4.5) V (t, (X(t) − X ∗ )) = ||(X(t) − X∗ )||22 , 2 where X = (x, y, z)T and X ∗ is an equilibrium point. Note that V is a positive definite function on R3 . From system (4.1), we have Dα V (t, (X(t) − X ∗ )) = −2a||(X(t) − X ∗ )||22 .

(4.6)

According to Theorem. (2.2), the equilibrium point E1 (resp. E2 and E3 ) is Mittag-Leffler stable, therefore it is asymptotically stable. Remark 4.1. For the intege-order case (α = 1), the direct method of Lyapunov is used to ensure the asymptotic stability of the solution of (4.1). The above Lyapunov fucntion (4.5) is equal to zero at the equilibrium of the system (4.1). Furthermore, the time derivative of V is V˙ (t, X(t)) = −2a||X(t)||2 ,

(4.7)

since a > 0 we have V˙ (t, X(t)) < 0, then according to Lasalle-Yoshizawa theorem the equilibrium E1 (0, 0, 0) of system (4.1) is globally asymptotically stable. 4.1 Numerical simulations As mentionned before, we have implemented the improved MsGDTM algorithm for numerical simulations. In order to compare between the behaviour of the trajectories without the control and with it, the activation of control is made when t ≥ 8. In what follows we describe the numerical technique for simulating the control process. • First we take α = 0.85 in order to ensure existence of chaos. The time range is [0, 3000], the time-step is 0.01 and the initial state is taken as (0.5, 0.2, 15). • Then, we run the MSGDTM code for system (4.1) without control from t = 1 to t = 800. • Finally, we add the active controllers and run the code from t = 801 to t = 3000. For α = 0.85, the unstable point E1 has been stabilized, as shown in Figures 4 and 5. From Figure 4, we see that the behaviour of x(t), y(t) and z(t) is chaotic, then when the control is started (t ≥ 8) the equilibrium point is rapidly stabilized. In Figure 5, we see that the trajectory converges to the equilibrium point E1 (0, 0, 0). The equilibria E2 and E3 are stabilized in the same manner (see Figures 6-9).

10

Z.Hammouch, T.Mekkaoui

(a)

(b)

(c)

Fig. 4: The trajectories of the controlled system (4.1) stabilizing equilibrium point E1 : (a) x(t), (b) y(t) and (c) z(t).

(d)

Fig. 5: The phase portrait x − y − z after activation of control towards E1 at tc = 8s.

(a)

(b)

(c)

Fig. 6: The trajectories of the controlled system (4.1) stabilizing equilibrium point E2 : (a) x(t), (b) y(t) and (c) z(t).

Control of a fractional-chaotic system by Mittag-Leffler stability

11

(d)

Fig. 7: The phase portrait x − y − z after activation of control towards E2 at tc = 8s.

(a)

(b)

(c)

Fig. 8: The trajectories of the controlled system (4.1) stabilizing equilibrium point E3 : (a) x(t), (b) y(t) and (c) z(t).

5 Conclusion In this paper, the active control strategy is considered. The chaos is controlled to unstable equilibrium effectively. The numerical simulations proved the performance of the control technique vis MSGDTM algorithm. In conclusion, the numerical and the analytical approaches presented in this paper are much easier to realize in many chaotic fractional nonautonomous systems.

12

Z.Hammouch, T.Mekkaoui

(d)

Fig. 9: The phase portrait x − y − z after activation of control towards E3 at tc = 8s.

Appendix

restart a := 38: b := 3: c := 30: d := 1: e := 1: alpha := .9: X[1,0]:=.5: Y[1,0]:=.2: Z[1,0]:=15: dt:= 0.01: G := k -> GAMMA(alpha*k+1)/GAMMA(alpha*(k+1)+1): for i to 3000 do for k from 0 to 4 do X[i,k+1]:=G(k)*a*(Y[i,k]-X[i,k]); Y[i,k+1]:=G(k)((c-a)*X[i,k]+c*Y[i,k] -d*(sum(X[i,r]*Z[i,k-r],r=0..k))); Z[i,k+1]:=G(k)*(-b*Z[i,k]+e*(sum(Y[i,r]*Y[i,k-r],r =0..k)); end do: X[i+1,0]:=sum(X[i,l]*(dt)ˆ(alpha*l),l=0..5): Y[i+1,0]:=sum(Y[i,l]*(dt)ˆ(alpha*l),l=0..5): Z[i+1,0]:=sum(Z[i,l]*(dt)ˆ(alpha*l),l=0..5): end do: for j to 3000 do f[j] := sum(X[j,m]*(t-(j-1)*dt)ˆ(alpha*m),m =0..5) end do: for l to 3000 do x[l]:= piecewise((l-1)*dt