Abstract
The dynamics of a discrete-time hyperchaotic system and the amplitude control of Hopf bifurcation for a stochastic discrete-time hyperchaotic system are investigated in this paper. Numerical simulations are presented to exhibit the complex dynamical behaviors in the discrete-time hyperchaotic system. Furthermore, the stochastic discrete-time hyperchaotic system with random parameters is transformed into its equivalent deterministic system with the orthogonal polynomial theory of discrete random function. In addition, the dynamical features of the discrete-time hyperchaotic system with random disturbances are obtained through its equivalent deterministic system. By using the Hopf bifurcation conditions of the deterministic discrete-time system, the specific conditions for the existence of Hopf bifurcation in the equivalent deterministic system are derived. And the amplitude control with random intensity is discussed in detail. Finally, the feasibility of the control method is demonstrated by numerical simulations.
Highlights
The dynamics and their bifurcation control in chaos systems have been given much attention and widely used in chemical and biological population and power systems [1, 2]
These results reveal far richer dynamics of the discrete model compared with the chaos system
Numerical simulations are presented to illustrate our results with the theoretical analysis and to exhibit the complex dynamical behaviors
Summary
The dynamics and their bifurcation control in chaos systems have been given much attention and widely used in chemical and biological population and power systems [1, 2]. The dynamical behaviors via the improved method in stochastic systems with random parameters are studied by Leng et al [21]. Hopf bifurcation control for stochastic dynamical system with nonlinear random feedback method has been investigated [25]. The dynamics analysis of stochastic discrete-time hyperchaotic system with random parameter, which is scarcely investigated, attracts our interest. 2. Orthogonal Polynomial Approximation of a Discrete-Time Hyperchaotic System with Random Parameter. In order to facilitate the numerical analysis of this paper, we select M = 1 and γ = 1 and approximately obtain the equivalent deterministic system of discrete-time hyperchaotic system with random parameter: x0 (n + 1) = ay0 (n) + S0 (n) + U0 (n) , y0 (n + 1) = dx0 (n) + δx0 (n) , (6).
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