Construction of a New Fractional Chaotic System and Generalized Synchronization

Commun. Theor. Phys. (Beijing, China) 5 (2010) pp. 1105 1110 c Chinese Physical Society and IOP Publishing Ltd Vol. 5, No. 6, June 15, 2010 Construction of a New Fractional Chaotic System and Generalized Synchronization ZHANG Xiao-Dan ( ), 1, ZHAO Pin-Dong ( ), 1, and LI Ai-Hua (ÓÇÙ) 2, 1 School of Applied Science, University of Science and Technology Beijing, Beijing 10008, China 2 Department of Mathematical Science, Montclair State University, New Jersey 0704, USA (Received November 1, 2009; revised manuscript received January 1, 2010) Abstract In this paper a new dynamic system with integer and fractional order is investigated. It is shown that determining the effect of quadratic coefficients to the systematic structure can be converted to determining that of coefficients of the linear part. Under some parametric conditions, the system can produce chaotic attractors similar as Lorenz attractor. A constructive theorem is proposed for generalized synchronization related to the fractional-order chaotic system and an application of this new system is demonstrated. PACS numbers: 05.45.Gg Key words: chaos, fractional-order system, generalized synchronization 1 Introduction The ideas of fractional integrals and derivatives have been known since the development of the regular calculus. Although fractional calculus has a 00-year-old history, its applications to physics and engineering are just presenting a recent focus of interest. It was found that many dynamical systems in interdisciplinary fields can be elegantly described with the help of fractional derivatives. For instance, many systems are known that can display fractional-order dynamics, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, electromagnetic waves, quantitative finance, and quantum evolution of complex systems. [1 6] Furthermore, some systems have been found that have chaotic motions in the fractional order. The behavior of nonlinear chaotic systems with fractional models was also investigated widely and reported. [2 7] According to the Poincaré Bendixon Theorem, [8] chaos cannot occur in autonomous continuous-time systems of integer-order less than three. The Chen system [9] is a recent example of a continuous-time system of order three that exhibits chaos. The order of this system is defined as the sum of the orders of all involved derivatives. However, in autonomous fractional-order systems, it is not the case. For example, it has been shown that the fractional-order Lorenz system with order as low as 2.82 could produce a chaotic attractor. [10] Chaos and hyperchaos in fractionalorder Rossler equations are also discussed in [11] where it is shown that chaos can exist in the fractional-order Rossler equation with order as low as 2.4 and hyper chaos can exist in the fractional-order Rossler hyper chaotic system with order as low as.8. Many recent investigations have been devoted to chaotic dynamics of fractional order. In [2, 12], chaotic behaviors in the fractional-order Chen system are studied and it is shown that the lowest order to have chaos in this system is 2.1 and 2.92 respectively. In recent years, synchronization of fractional-order chaotic system is also investigated. [1 15] In this paper, we investigate the chaotic behavior of a new integer and fractional-order system and found the lowest order for chaos to exist in the fractional-order system by performing numerical simulations. Finally, applications of this system and a simple generalize synchronization method are presented. 2 Dynamical System of Fractional Order There are many definitions of fractional derivatives. [1] Perhaps the best-known is the Riemann Liouvile definition. The Riemann Liouville derivative of order α is defined as d n adt α f(t) = 1 Γ(n α) dt n t a f(τ) dτ, (1) (t τ) α n+1 where Γ is the gamma function and n is an integer with n 1 < α < n. An alternative definition was introduced by Caputo as follows, which is a sort of regularization of the Riemann Liouville derivative: ad α t f(t) = Supported by the National Nature Science Foundation of China under Grant No. 60674059 E-mail: bkdzxd@16.com E-mail: zhaopd@126.com E-mail: Lia@mail.montclair.edu 1 Γ(n α) t a f (n) (x) dx. (2) (t x) α+1 n The main advantage of the definition given by Eq. (2) is that the Caputo derivative of a constant is equal to zero. Consider the following commensurate fractional order system d α x = f(x), () dtα

1106 ZHANG Xiao-Dan, ZHAO Pin-Dong, and LI Ai-Hua Vol. 5 where 0 < α 1 and x R n. The equilibrium points of system () are calculated by solving the equation f(x) = 0. Analyzed in [16], these equilibrium points are locally asymptotically stable if all the eigenvalues of the Jacobian matrix A = f/ x evaluated at them satisfy the following condition arg(eig(a)) > απ/2, (4) The inequality (4) shows that fractional order differential equations are more stable than their integer-order counterpart, which is quite reasonable because the systems with more memory are typically more stable than their memory-less counterpart. [16] Both stable regions and unstable regions are shown in Fig. 1. Fig. 1 Stability region of linear fractional order system with order απ/2. Construction and Analysis Now let us introduce a three-dimensional quadratic autonomous system: d α1 x dt = a 1x + a 2 y + d 1 xy, α1 d α2 y dt = by + d 2xz, α2 d α z dt = c 1y c 2 z + d xy, (5) α where 0 < α i 1 (i = 1, 2, ) and the system parameters are all real with d 1, d 2 < 0, d > 0 and a 1, a 2, b, c 1, c 2 R. The time derivatives refer to the Caputo derivatives. The effective order of the system (5) is defined as α 1 +α 2 +α. By performing the linear transformation x y z = ±A X Y Z, where ( 1 1 d ) A = diag,,, d2 d d 1 d 1 d2 d we obtain d α1 X = a 1 X + a 2 d2 d Y XY, dt α1 d 1 d α2 Y = by XY, dt α2 d α Z = c 1 d2 d Y c 2 Z ± XY. dt α d It implies that the effect of quadratic coefficients to the structure of system (5) can be altered to the effect of coefficients of linear terms. That is, investigating the effect of quadratic coefficients to system (5) can be converted to investigating the following system: d α1 x dt = a d α2 y 1x + a 2 y xy, = by xz, α1 α2 dt d α z dt = c 1y c 2 z + xy, (6) α when N = c 2 1 + 4bc 2 > 0. System (6) has three equilibriums: E 1 (0, 0, 0), E 2 (x 1, y 1, z 1 ), and E (x 2, y 2, z 2 ), where x 1 = 1 2 ( c 1 + N), y 1 = a 1( c 1 + N) 2a 2 ( c 1 + N), 2a 1 b z 1 = 2a 2 ( c 1 + N), x 2 = 1 2 ( c 1 N), y 2 = a 1( c 1 N) 2a 2 + (c 1 + N), z 2a 1 b 2 = 2a 2 + (c 1 + N). Let (x eq, y eq, z eq ) be any equilibrium point of system (6). Then the Jacobian matrix of system (6) at (x eq, y eq, z eq ) is given by a 1 y eq a 2 x eq 0 J = z eq b x eq. y eq c 1 + x eq c 2 The characteristic polynomial of J is where 2p 1 f(λ) = λi J = λ + p 1 λ 2 + p 2 λ + p, (7) p 1 = b + c 2 + a 1 y eq, p 2 = x 2 eq + (c 1 z eq )x eq + ( b + c 2 )y eq bc 2 + a 2 z eq + a 1 c 2 a 1 b, p = a 1 x 2 eq + ((c 1 + a 2 )y eq + a 1 c 1 c 2 z eq )x eq a 1 bc 2 + a 2 c 2 z eq bc 2 y eq. Let λ = µ p 1 /, p = p 2 p 2 1/, and q = p p 1 p 2 /+ /27, then let f(λ) = 0 can be simplified as µ + pµ + q = 0. (8) Furthermore, using Cardano s formula, if the discriminant = q 2 /4 + p /27 > 0, then Eq. (8) has a unique real root: λ 1 = p 1 + q2 + + q 2, along with two complex conjugate roots: λ 2, = p 1 1 ( q2 2 + + q 2 )

No. 6 Construction of a New Fractional Chaotic System and Generalized Synchronization 1107 ± ( 2 i q2 + + q 2 ). Obviously, if the parameters satisfy the following conditions: > 0, q > 0, q2 + + q 2 < p 1 < 0, then λ 1 < 0, Re(λ 2, ) > 0, and λ 1 > Re(λ 2, ). Thus the equilibriums of system (6) satisfy the conditions of Shil nikov Theorem. [17] 4 Numerical Simulation When a 1 = a 2 = 7, b = 27, c 2 = 7, the non-original equilibriums and its corresponding eigenvalues of system (6) are given in Table 1. The two non-original equilibriums E 2 and E are not symmetric with respect to z-axis and the eigenvalues of the Jacobian matrix evaluated at these two equilibriums are different. The asymmetric relation generates the asymmetrical structure of chaotic attractor which is different from Lorenz system, [18] Chen system, [19] Lü system, [20] and some others. Table 1 The non-original equilibrium points and corresponding eigenvalues of system (6). equilibriums eigenvalues c 1 = 10 E 2 = (9.6, 1.02, 6.50) λ 1 = 7.47, λ 2, =.7 ± 16.26i E = ( 19.6, 12.8, 17.64) λ 1 = 22.70, λ 2, = 9.26 ± 29.16i Fig. 2 The chaotic attractor of system (6) at a 1 = a 2 = 7, b = 27, c 1 = 10, c 2 = 7, α i = 1 (i = 1,2, ). Thus in the case α i = 14 (i = 1, 2, ), system (6) is reduced to a common dynamical system exhibiting chaotic behavior. Using the algorithm in [21], we claim that the max Lyaunov exponent is 6.5. The trajectories for system (6) are shown in Fig. 2 when α i = 1 (i = 1, 2, ). When α 1 = α 2 = α < 1, a 1 = a 2 = 7, b = 27, c 2 = 7, c 1 = 10, system (6) does not always remain chaotic when changing the values of α i < 1 (i = 1, 2, ). From inequality (4), the necessary condition for system (6) to remain chaotic is keeping the two conjugate eigenvalues of the Jacobian matrix evaluated at each non-original equilibrium point in the unstable region. [22] This means that α i > (2/π)max{arctan(16.26/.7), arctan(29.16/9.26)} 0.8564, for i = 1, 2, and c 1 = 10.

1108 ZHANG Xiao-Dan, ZHAO Pin-Dong, and LI Ai-Hua Vol. 5 By using predictor-corrector scheme, [11,2 24] the trajectories for system (6) are described and shown in Fig. when α i = 0.95 (i = 1, 2, ). Fig. The chaotic attractor of system (6) at a 1 = a 2 = 7, b = 27, c 1 = 10, c 2 = 7, α i = 0.95 (i = 1, 2, ). 5 Generalized Synchronization for Corresponding Chaotic System of Fractional Order Based on the results in [25 26], a simple synchronization method is presented for the chaotic systems with fractional order mentioned above. Define 1 Consider two systems of fractional order d α x = F(x), dtα (9) d α y = G(x), dtα (10) where x R n, y R m, F(x) = (f 1 (x), f 2 (x),..., f n (x)) T R n, and G(y, x) = (g 1 (y, x), g 2 (y, x),..., g m (y, x)) T R m. If there exists a transformation H : R n R m and a manifold M = {(x, y) : y = H(x)} contained in a set B = B x B y R n R m such that all trajectories of Eqs. (9) and (10) with initial conditions in B approach M as t, i.e., lim n y H(x) = 0, then systems (9) and (10) are said to be in Generalized Synchronization (GS) with respect to the transformation H. The two systems are also called undirectionally coupled GS systems. Theorem 1 Let A, M, K R n n, where M is an invertible matrix. If every eigenvalue λ of the matrix (MAM 1 + K) satisfies arg(λ) > απ/2, then the following systems (11) and (12) are in GS via the linear transformation y = Mx: d α x = Ax + ϕ(x), dtα (11) d α y dt α = MAM 1 y + Mϕ(x) + Ky KMx. (12) Proof Let the error be e = y Mx. Then the error system is given by d α e dt α = dα y dt α M dα x dt α = MAM 1 y + Mϕ(x) + Ky KMx M(Ax + ϕ(x)) = MAM 1 y MAx + K(y Mx) = MAM 1 (y Mx) + K(y Mx) = (MAM 1 + K)(y Mx) = (MAM 1 + K)e. Since every eigenvalue λ of the matrix (MAM 1 +K) satisfies arg(λ) > απ/2, e is locally asymptotically sta-

No. 6 Construction of a New Fractional Chaotic System and Generalized Synchronization 1109 ble. It follows that the two dynamic systems described in (11) and (12) are in GS via transformation y = Mx. Example Let the driving system be system (6) where a 1 = a 2 = 7, b = 27, c 1 = 10, c 2 = 7, α i = 0.95 (i = 1, 2, ), and a 1 a 2 0 xy A = 0 b 0, ϕ = xz, c 1 c 2 0 xy 2 0 0 0 0 M = 0 4 0, K = 0 7 0, 1 0 2 0 0 0 then every eigenvalue λ of matrix (MAM 1 +K) satisfies the inequality arg(λ) > απ/2. By Theorem 1, the driven system d α x dt α = 7x + 66.5y 2xy xz, d α y dt α = 10y + 148y 4xz, d α z dt α = 15x + 15.5y 7z xy, is GS via the transformation (x, y, z ) T = M(x, y, z) T. The computer simulation result is shown in Fig. 4. The trajectories of the driven system are exactly GS with respect to the chaotic trajectories of the driving system. Fig. 4 (a) The trajectories of the driving system are chaotic. (b) The trajectories of the driving system exhibit GS. (c) The trajectories given in (a) are decoded, which show the same patterns as those given in (b). (d) Variables x and y are in GS via the transformation M. 6 Conclusions This paper constructed a new dynamical system with integral and fractional order and developed a method for designing predictable GS systems. Numerical simulations show that the system can produce chaotic attractors in a large parameter range. The efficiency of our GS method is demonstrated by the examples given in the paper. References [1] I. Podlubny, Fractional Differential Equations, Academic, New York (1999). [2] C. Li and G.R. Chen, Chaos, Solitions and Fractals 22 (2004) 549. [] W.M. Ahmad and A.M. Harb, Chaos, Solitions and Fractals 18 (200) 69. [4] W.M. Ahmad, E.K. Reyad, and A.A. Yousef, Chaos, Solitions and Fractals 22 (2004) 141. [5] W.M. Ahmad, Chaos, Solitions and Fractals 26 (2005) 1459.

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