Generalized Synchronization of Different Dimensional Chaotic Systems Based on Parameter Identification

Abstract

This paper investigates the generalized synchronization issue for two different dimensional chaotic systems with unknown parameters. Based on Lyapunov stability theory and adaptive control theory, an adaptive controller is derived to achieve the generalized synchronization whether the dimension of drive system is greater than the one of the response system or not. Meanwhile, corresponding parameter updating laws can be obtained so as to exactly identify uncertain parameters. This technique has been successfully applied to two examples, the generalized synchronization of hyperchaotic Rössler system and chaotic Lorenz system, chaotic Chen system and generalized Lorenz system. Numerical simulations are finally shown to illustrate the effectiveness of the proposed approach.

Quadratic optimal synchronization of uncertain chaotic systems

“…Hence, if a matrix P can be found that satisfies (9) "t 2 (t 0 , 1), then the value function V(e, t) given by (12) satisfies the H-J-B equation in (11). In this case, the optimal control law u à o ðtÞ is obtained as (10). Moreover, since the quadratic function V(e, t) in (12) is positive, it can be chosen as a Lyapunov function candidate.…”

“…Consider the master and slave uncertain chaotic systems described in (19) and (20) with the robust control law u(t) given by (26) , where the optimal input u o (t) is specified as (10) . If the parameterâ is tuned by (29) , and there exist positive definite matrices Q = Q T and R = R T such that…”

“…In practical situation, some parameters of chaotic systems are probably unknown and may change over time. Thus, adaptive control of chaotic systems is motivated by applications in master-slave synchronization [9][10][11]. Yang [12] addressed robust adaptive synchronization problem of chaotic systems with uncertainties.…”

A Robust Control Method forQ-SSynchronization between Different Dimensional Integer-Order and Fractional-Order Chaotic Systems

“…Also, complete synchronization and antisynchronization have been observed, for example, in [16][17][18], and function projective synchronization has been studied in [19]. Until now, a variety of control schemes have been proposed to study the problem of chaos synchronization between different dimensional systems such as modified function projective synchronization [20], generalized matrix projective synchronization [21], generalized synchronization [22][23][24], inverse generalized synchronization [25], full state hybrid projective synchronization [26], Q-S synchronization [27], increased order synchronization [28,29], and reduced order generalized synchronization [30]. Amongst all kinds of synchronization, Q-S synchronization has been extensively considered [31][32][33][34][35][36][37][38][39], due to its universality and its great potential applications in applied sciences and engineering.…”

Robust Synchronization of Hyperchaotic Systems with Uncertainties and External Disturbances

“…The idea of synchronization is to use the output of the drive system to control the response system so that the output of the response system follows the one of the drive system asymptotically. Up to now, many types of synchronization phenomena have been reported, such as generalized synchronization [4,5], adaptive synchronization [6][7][8][9], projective synchronization [10][11][12], impulsive synchronization [13], lag synchronization [14,15], and function projective synchronization [16,17]. And a wide variety of control approaches, such as backstepping design technique [18], fuzzy sliding mode control [19,20], adaptive control [21], optimal control [22], and ∞ control [23], have been proposed to synchronize chaotic systems.…”