Active Control and Global Synchronization of the Complex Chen and Lü Systems

Abstract

Chaos synchronization is a very important nonlinear phenomenon, which has been studied to date extensively on dynamical systems described by real variables. There also exist, however, interesting cases of dynamical systems, where the main variables participating in the dynamics are complex, for example, when amplitudes of electromagnetic fields are involved. Another example is when chaos synchronization is used for communications, where doubling the number of variables may be used to increase the content and security of the transmitted information. It is also well-known that similar generalization of the Lorenz system to one with complex ODEs has been introduced to describe and simulate the physics of a detuned laser and thermal convection of liquid flows. In this paper, we study chaos synchronization by applying active control and Lyapunov function analysis to two such systems introduced by Chen and Lü. First we show that, written in terms of complex variables, these systems can have chaotic dynamics and exhibit strange attractors. We calculate numerically the values of the parameters at which these attractors exist. Active control and global synchronization techniques are then applied to study the phenomenon of chaos synchronization. Analytical criteria concerning the stability of these techniques are implemented and excellent agreement is found upon comparison with numerical experiments. In particular, studying the time evolution of "errors" (or differences between drive and control dynamics), we show that both techniques are very effective for controlling the behavior of these systems, even in regimes of very strong chaos.

Combination Synchronization of Three Identical or Different Nonlinear Complex Hyperchaotic Systems

“…The system parameters are given as α = 8, β = 5, γ = 50 and σ = 15, so that the complex Lorenz system exhibits hyperchaotic behavior. We assume Figure 1 displays the time response of the combination synchronization errors, e 1 , e 2 , e 3 , e 4 , e 5 ande 6 . The errors converge to zero, which implies that systems (5), (6) and (7) (7) with controller (18) are displayed in Figure 3, which illustrates that system (7) is stabilized to the equilibrium, O(0, 0, 0, 0, 0, 0).…”

“…In [3], the authors studied the chaotic unstable limit cycles of complex Van der Pol oscillators. The rich dynamics behaviors of the complex Chen and complex Lü systems were investigated in [4]. By adding state feedback controllers to their complex chaotic systems, complex hyperchaotic Chen, Lorenz and Lü systems were introduced and studied in [5][6][7], respectively.…”

Modified Projective Synchronization of Chaotic Systems with Noise Disturbance, an Active Nonlinear Control Method

“…Up to now, many types of controling methods are revealed and investigated for control and synchronization of chaotic systems. Active method [2,3,4,5,6], adaptive method [7,8,9], linear feedback method [10,11], nonlinear feedback method [12,14,15], sliding mode method [16,17,18], impulsive method [19], phase method [20], generalized method [21], robust synchronization [13] and projective method [22,23,24] are some of the introduced methods by the researchers. Among these methods, synchronization with some types of projective methods are extensively investigated in the last decades, since the faster synchronization due to its synchronization scaling factors, which master and slave chaotic systems would be synchronized up to a proportional rate.…”

Switched modified function projective synchronization between two complex nonlinear hyperchaotic systems based on adaptive control and parameter identification

“…[18], the authors studied chaotic unstable limit cycles of complex Van der Pol oscillators. The rich dynamical behaviors of the complex Chen and complex Lü systems were investigated in [19]. By adding state feedback controllers to their complex chaotic systems, complex hyperchaotic Chen, Lorenz, Lü systems were introduced and studied in [20] - [22], respectively.…”