Generalized synchronization is a typical dynamical phenomenon in nonlinear systems, for which the real-valued setting has been widely investigated. The complex-valued functions relationship in generalized synchronization is equally important for complex-valued dynamical systems, which however are seldom studied. Complex parameters identification on the synchronization manifold remains an open problem owing to the absence of the persistent excitation (PE) condition in the complex field. This paper investigates generalized synchronization via a complex-valued vector mapping (CGS) for different-dimensional complex-variable chaotic (hyper-chaotic) systems (CVCSs) with complex parameters identification. Based on Lyapunov stability theory in the complex field and using an adaptive control method, some sufficient criteria are established to achieve CGS for CVCSs. Moreover, some necessary and sufficient criteria are derived to ensure complex parameters identification. Finally, the theoretical results are verified and demonstrated by reduced-order and increased-order simulation examples.
Chaotic dynamics analysis of complex-variable chaotic systems (CVCSs) is an important problem in real secure communication and encryption. In this paper, a simple one-parameter chaotic system in complex field is proposed, whose nonlinear terms are the same as Lorenz system but the linear terms are much simpler. The proposed system has circular equilibria and therefore multi-stability can be measured by phase portraits, bifurcation diagrams and Lyapunov exponent spectrum. Its basin of attraction is filled with initial points leading to chaotic behaviors. The coexistence of infinitely many attractors is found in the proposed system, which is not reported in the existing complex-variable Lorenz system. Finally, two complexity indexes are used to measure dynamic characteristic with respect to parameter. INDEX TERMS Coexisting attractors, extreme multistability, complex-variable chaotic system, complexity.
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