Combination complex synchronization of three chaotic complex systems

Abstract

In this paper, we firstly design a chaotic complex system and study its dynamical properties including invariance, dissipativity, equilibria, Lyapunov exponents, chaotic behavior, as well as chaotic attractors. What is more, the scaling matrices are always chosen as real matrices in previous combination synchronization schemes within two drive real systems and one response real system evolving in the same or inverse directions simultaneously. However, in many real-life applications, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, combination synchronization with regard to the complex scaling matrices, referred as combination complex synchronization, will be made the further research about three chaotic complex systems. Based on Lyapunov stability theory, three identical chaotic complex systems are considered and the corresponding controllers are designed to achieve the complex combination synchronization. The corresponding theoretical proofs and numerical simulations are given to demonstrate the validity and feasibility of the presented control technique.

Fixed-Time Complex Modified Function Projective Lag Synchronization of Chaotic (Hyperchaotic) Complex Systems

“…Since expression (16) implies that ( ) is decreasing, (18) indicates that ( ) will become zero in finite time.…”

“…Recently, some new types of synchronization that utilized complex scaling factors have been introduced to achieve the synchronization of both module and phase, the concept first proposed by Nian et al [14]. Complex complete synchronization (CCoS) [15,16], complex projective synchronization (CPS) [17], combination complex synchronization [18], complex modified projective synchronization (CMPS) [19], and complex function projective synchronization (CFPS) [20,21] are some examples. In these synchronizations, the scaling factors are complex numbers (or functions) that increase the unpredictability and complexity; they have thus significant meaning for enhancing the security of communication [20].…”

“…In order to obtain such complex synchronization of chaotic (hyperchaotic) complex systems, some control methods such as Lyapunov-based control, feedback control and/or adaptive control [2,5,6,8,[11][12][13][14][15][17][18][19][20], passive control [3], finite-time stability theory-based control [4,9,16], nonlinear observer-based control [7], backstepping-based control [10], and adaptive fuzzy logic control [21] have been proposed. In the mentioned methods, while the ones in [4,9,16] ensure obtaining finite-time synchronization in which the convergence time still depends on both control design parameters 2 Complexity and initial conditions, these remainders guarantee that the synchronization errors might converge to origin asymptotically in the best case.…”

Adaptive Modified Function Projective Synchronization of Uncertain Complex Dynamical Networks with Multiple Time-Delay Couplings and Disturbances

“…Synchronization is a fundamental phenomenon in nature which enables the networks to achieve coherent behavior due to interaction. The research of complex network synchronization is one of the most important research directions on complex networks [6][7][8]. Up to now, many types of synchronization phenomena on complex networks have been reported, such as complete synchronization [9,10], projective synchronization [11,12], lag synchronization [13], lag projective synchronization [14], bounded synchronization [3], projective cluster synchronization [15], and global synchronization [16].…”

Adaptive generalized combination complex synchronization of uncertain real and complex nonlinear systems

“…21 Sun et al investigated combination complex synchronization of three identical hyperchaotic complex systems via two complex scaling diagonal matrices. 22 Noticeably, Jiang et al proposed combination complex synchronization of chaotic real systems and complex systems with different dimensions via two complex scaling matrices. 23 As introduced in Refs.…”