Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters
Introduction
Since Rössler [1] first introduced the hyperchaotic dynamical system in 1979, many hyperchaotic systems have been proposed and studied in the last few decades, such as hyperchaotic Lorenz system [2], hyperchaotic Chen system [3], hyperchaotic Lü system [4], just to name a few. Hyperchaotic systems possess more complex dynamical behaviors than chaotic systems such as having more than one positive Lyapunov exponent, therefore, they have broader potential applications, particularly in secure communications.
Chaos synchronization plays an important role for understanding the cooperative behavior in coupled chaotic oscillators [5]. Since Pecora and Carroll [6] introduced a method to synchronize two identical chaotic systems with different initial conditions, chaos synchronization has attracted a great deal of attention from various fields during the last two decades. A variety of approaches have been proposed for the synchronization of chaotic and hyperchaotic systems such as linear and nonlinear feedback synchronization methods [7], [8], adaptive synchronization methods [9], [10], backstepping design methods [11], [12], and sliding mode control methods [13], etc. However, to our best knowledge, most of the methods mentioned above and many other existing synchronization methods mainly concern the synchronization of two identical chaotic or hyperchaotic systems, the methods of synchronization of two different chaotic or hyperchaotic systems are far from being straightforward because of their different structures and parameter mismatch. Moreover, most of the methods synchronize only two systems with exactly knowing of their structure and parameters. But in practical situations, some or all of the systems’ parameters cannot be exactly known in priori. As a result, more and more applications of chaos synchronization in secure communication have made it much more important to synchronize two different hyperchaotic systems with uncertain parameters in recent years. In this regard, some works on synchronization of two different hyperchaotic systems with uncertain parameters have been performed [14], [15], [16].
Recently, Chen et al. [17] proposed a novel hyperchaotic system by introducing state feedback control and constant multipliers to the two quadratic terms in the system reported in [18]. This novel hyperchaotic system takes the following form:where x; y; z and w are state variables, and a; b; c; d; e and g are constant parameters. System (1) has only one unstable equilibrium O(0, 0, 0, 0) and has bigger positive Lyapunov exponents than those already known hyperchaotic systems. It can generate complex dynamics within wide parameter ranges, including periodic orbit, quasi-periodic orbit, chaos and hyperchaos. In particular, when a = 35; b = 4.9; c = 25; d = 5; e = 35 and varying g from 10 to 126, or a = 35; c = 25; d = 5; e = 35; g = 100 and ranging b between 3.8 and 11, system (1) exhibits hyperchaos. This hyperchaotic attractor is given in Fig. 1.
Very recently, Wang et al. [19] generated another new hyperchaotic system from Lorenz system. The new hyperchaotic system is described bywhere h, k, p and r are constant parameters. When h = 10, k = 8/3, p = 28 and r ∈ (−1.52, −0.06), system (2) has two positive Lyapunov exponents λ1 = 0. 3381 and λ2 = 0. 1586. Thus, system (2) shows hyperchaotic behavior. This hyperchaotic attractor is given in Fig. 2.
To our knowledge, synchronizing hyperchaotic systems (1), (2) with uncertain parameters has not been reported. This paper presents the different structure synchronization between these two novel hyperchaotic systems aforementioned with partly uncertain parameters. On the basis of the Lyapunov stability theory, we design a new adaptive synchronization controller with a novel parameter update law. With this adaptive controller, one can synchronize the two hyperchaotic systems effectively and identify the system parameters accurately.
The rest of the paper is organized as follows. Section 2 presents hyperchaos synchronization between hyperchaotic system (1), (2) via adaptive control. Section 3 provides a numerical example to demonstrate the effectiveness of the proposed method. Section 4 concludes the paper.
Section snippets
Adaptive synchronization
In order to observe synchronization behavior between the two hyperchaotic systems, we assume that the hyperchaotic system (1) is the drive system whose variables are denoted by subscript 1 and the parameters b and g cannot be exactly known in priori. The drive systems is described by the following equation:The hyperchaotic system (2) is the response system whose variables are denoted by subscript 2 and the parameter r cannot be
Numerical simulation
In this section, Numerical simulation is given to show the effectiveness and feasibility of the proposed synchronization method. In the numerical simulation, the Runge–Kutta–Fehlberg algorithm with adaptive varied time step is used to solve the augment system consisting of (3), (7), (8). The parameters are chosen to be a = 35, b = 5, c = 25, d = 5, e = 35, g = 10, h = 10, k = 8/3, p = 28 and r = −1 in the simulation so that system (3), (4) exhibit hyperchaotic behaviors if no controls are applied. The initial
Conclusion
This paper addresses the problem of adaptive synchronization of two different new hyperchaotic systems with some uncertain parameters. On the basis of the Lyapunov stability theory and the adaptive control theory, a new adaptive synchronization control law and a novel parameter estimation update law are proposed to achieve synchronization between the two novel different hyperchaotic systems with uncertain parameters. Numerical simulations are given to demonstrate the effectiveness of the
Acknowledgements
This work was supported by the Chinese Provincial Natural Science Foundation of Hunan province (No: 06JJ5098).
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