Projective synchronization of chaotic time-delayed systems via sliding mode controller

Further results on the impulsive synchronization of uncertain complex‐variable chaotic delayed systems

“…where p; q, and d à are defined to be the same as Theorem 1. There exist x k 5maxð1; l k 1t k exp k à s Þ and k à 5 inf t!t0 kðtÞ > 0; kðtÞ2p1qexp kðtÞs 50g; È where l k 5maxf 1; 2kI1B k k 2 g; t k 5maxf1; 2kC k k 2 g: Then the response chaotic delayed complex-variable system (4) can synchronize the drive chaotic delayed complex-variable system (3) asymptotically with the impulsive controller (6), the adaptive controller (10), and the parameter update law (11). Moreover,Ĥ !…”

“…0; a k 5k max ðI1A k Þ T ðI1A k Þ < 1; 2 kI1B k k 2 ! a k : There exist a constant d > 1 satisfying lnðg exp ks Þ < dks such that inf k2N ðt k 2t k21 Þ > ds; where g5 sup k2N 1; 2kI1B k k 2 12kC k k 2 exp ks g; n and k > 0 is the unique solution of equation k5ð2d à 2lÞ2kexp ks : Then the response chaotic delayed complex-variable system (4) can synchronize the drive chaotic delayed complex-variable system (3) asymptotically with the impulsive controller (6), the adaptive controller (10), and the parameter update law (11). Moreover,Ĥ !…”

“…Simultaneously, some kinds of synchronization have been intensively investigated and a lot of theoretical results have been obtained during the last two decades, such as complete synchronization, antisynchronization, generalized synchronization, phase synchronization, antiphase synchronization, lag synchronization, partial synchronization, projective synchronization, combination synchronization, and so forth. At the same time, several theoretical methods have been developed to realize chaos synchronization such as feedback control method [3,4], active control method [5,6], backstepping method [7], adaptive control method [8][9][10], sliding mode control method [11], impulsive control method [12,13], and coupling control method [14,15].…”

Projective lag synchronization of Markovian jumping neural networks with mode-dependent mixed time-delays based on an integral sliding mode controller

“…A large portion of the vast literature on the neural networks with sliding modes application is dealing with the synchronization of neural networks [47][48][49][50]. Sliding mode control (SMC) methods can be considered as a synthesis procedure, which is often associated with the theory of variable structure control.…”

Adaptive‐impulsive function projective synchronization for a class of time‐delay chaotic systems

“…Clearly, complete synchronization and antisynchronization are special cases of projective synchronization, respectively. In recent years, some researchers have paid their attention to investigate the projective synchronization for chaotic systems [17][18][19][20][21] and complex dynamical networks [22][23][24][25][26][27]. Later, a new synchronization method called modified projective synchronization (MPS) was proposed in [28], where the chaotic systems can synchronize up to a constant scaling matrix.…”