Chaos Control and Synchronization of a Novel Chaotic System Based upon Adaptive Control Algorithm

Abstract

Abstract-Controlling chaos is stabilizing one of the unstable periodic orbits either to its equilibrium point or to a stable periodic orbit by means of an appropriate continuous signal injected to the system. On the other hand, chaos synchronization refers to a procedure where two chaotic oscillators (either identical or nonidentical) adjust a given property of their motion to a common behavior. This research paper concerns itself with the Adaptive control and synchronization of a new chaotic system with unknown parameters. Based on the Lyapunov direct method (LDM), the Adaptive control techniques (ACTs) are designed in such a way that the trajectory of the new chaotic system is globally stabilized to one of its equilibrium points of the uncontrolled system. Moreover, the Adaptive control law is also applied to achieve the synchronization state of two identical systems and two different chaotic systems with fully unknown parameters. The parameters identification, chaos control and synchronization of the chaotic system have been carried out simultaneously by the Adaptive controller. All simulation results are carried out to corroborate the effectiveness and the robustness of the proposed methodology and possible feasibility for synchronizing two chaotic systems by using Mathematica 9.

A Research on Active Control to Synchronize a New 3D Chaotic System

“…For the identical chaotic systems [21], the time series of the state vectors of the synchronized and unsynchronized trajectories are depicted in Figures 2-4. The state trajectories of the drive system converged to the state trajectories of the response system under the control action Equation (12).…”

“…Various linear and nonlinear control techniques have been developed to carry out synchronization behavior. Some of these include, adaptive control [7], linear error state feedback control [8], backstepping method [9], active control [10], sliding mode control [11], and nonlinear control [12] are worth citing here, among others.…”

Analysis, dynamics and adaptive control synchronization of a novel chaotic 3-D system

“…In (15) and (16), the system parameters a, b, c are unknown and the design goal is to find an adaptive feedback controls u 1 , u 2 , u 3 using the states x 1 , x 2 , x 3 and estimates;…”

“…The synchronization error between the novel chaotic systems (15) and (16) Thus, the synchronization error dynamics is obtained as: We take the adaptive control law defined by:…”

“…Many 3-D chaotic systems have been discovered such as Rössler system [2], Chen system [3], Lü system [4], Liu-Chen system [5], Zhu system [6], Sprott system [7] , Vaidyanathan system [8], Elhadj [9]...etc. In recent years, many methods of synchronization for chaotic systems have been introduced and applied such as; OYG method [10], backstepping design method [11], sliding mode control [12], passive control [13], fuzzy control [14], nonlinear active control [15,16], projective synchronization [17], function projective synchronization [18], global synchronization [19,20]...etc. Adaptive control method is used when parameters are unknown or initially uncertain.…”