Abstract
In this paper, the complex simplified Lorenz system is proposed. It is the complex extension of the simplified Lorenz system. Dynamics of the proposed system is investigated by theoretical analysis as well as numerical simulation, including bifurcation diagram, Lyapunov exponent spectrum, phase portraits, Poincaré section, and basins of attraction. The results show that the complex simplified Lorenz system has non-trivial circular equilibria and displays abundant and complicated dynamical behaviors. Particularly, the coexistence of infinitely many attractors, i.e., extreme multistability, is discovered in the proposed system. Furthermore, the adaptive complex generalized function projective synchronization between two complex simplified Lorenz systems with unknown parameter is achieved. Based on Lyapunov stability theory, the corresponding adaptive controllers and parameter update law are designed. The numerical simulation results demonstrate the effectiveness and feasibility of the proposed synchronization scheme. It provides a theoretical and experimental basis for the applications of the complex simplified Lorenz system.
Highlights
Since Lorenz discovered the first chaotic attractor in 1963 [1], chaos has been extensively investigated overApart from the above-mentioned complex Lorenz system, some other complex chaotic systems have been proposed and investigated theoretically and numerically, such as the complex Chen system [13], complex Lü system [13], hyperchaotic complex Lorenz system [14] and hyperchaotic complex Lü system [15]
The results show that the complex simplified Lorenz system has non-trivial circular equilibria and displays abundant and complicated dynamical behaviors
Some scholars focused their attention on the study of chaos control and synchronization of complex chaotic systems in recent years, and a great number of synchronization schemes have been proposed, such as complete synchronization [16], lag synchronization [17], complex modified projective synchronization [18], adaptive impulsive synchronization [19], adaptive complex modified projective synchronization [20], combination complex synchronization [21], adaptive complex modified hybrid function projective synchronization [22], and adaptive complex modified function projective synchronization [23]
Summary
Since Lorenz discovered the first chaotic attractor in 1963 [1], chaos has been extensively investigated over. The extreme multistability is an intrinsic property of many nonlinear dynamical systems It means the coexistence of infinitely many asymptotic stable states for a given set of parameters. Motivated by the above discussion, in this paper, by extending the real state variables from real domain to complex domain, the complex simplified Lorenz system is proposed. The adaptive complex generalized function projective synchronization between two complex simplified Lorenz systems with unknown parameter is achieved.
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