新四维超混沌系统的复杂动力学研究

Research on Complex Dynamics of a New 4D Hyperchaotic System

本文基于三维Lorenz-like混沌系统,设计线性反馈控制器,提出了一个仅有2个二次非线性项的新四维超混沌系统。此系统具有简单的代数结构,但却展现复杂的动力学行为,并理论证明它与超混沌Li系统是不等价的。为了研究系统的复杂动力学,本文详细探讨了系统在双曲和非双曲平衡点时的稳定性,且严格分析Hopf分岔,获得Hopf分岔所产生周期轨的近似表达式和稳定性。进一步借助现代数学软件进行数值仿真,得到系统的Lyapunov指数谱、Poincaré映射和分岔图,验证系统超混沌吸引子的存在性。

In this paper,based on the 3D Lorenz-like chaotic system,a linear feedback controller is designed and a new four-dimensional hyperchaos system with only two times nonlinear terms is proposed.This system has simple algebraic structure,but shows complex dynamic behavior,and it is proved theoretically that it is not equivalent to hyperchaotic Li system.In order to study the complex dynamics of the system,the stability of the system at the hyperbolic and non-hyperbolic equilibrium points is discussed in detail,and the Hopf bifurcation is strictly analyzed.The approximate expression and stability of the periodic orbit generated by the Hopf bifurcation are obtained.Furthermore,the Lyapunov exponent spectrum,Poincarémap and bifurcation diagram of the system are obtained by numerical simulation with the help of modern mathematical software,and the existence of the hyperchaotic attractor is verified.