Zhang Sen, Zeng Yicheng, Li Zhijun, Wang Mengjiao, Xiong Le
School of Physics and Opotoelectric Engineering, Xiangtan University, Xiangtan, Hunan 411105, China.
College of Information Engineering, Xiangtan University, Xiangtan, Hunan 411105, China.
Chaos. 2018 Jan;28(1):013113. doi: 10.1063/1.5006214.
By using a simple state feedback controller in a three-dimensional chaotic system, a novel 4D chaotic system is derived in this paper. The system state equations are composed of nine terms including only one constant term. Depending on the different values of the constant term, this new proposed system has a line of equilibrium points or no equilibrium points. Compared with other similar chaotic systems, the newly presented system owns more abundant and complicated dynamic properties. What interests us is the observation that if the value of the constant term of the system is nonzero, it has no equilibria, and therefore, the Shil'nikov theorem is not suitable to verify the existence of chaos for the lack of heteroclinic or homoclinic trajectory. However, one-wing, two-wing, three-wing, and four-wing hidden attractors can be obtained from this new system. In addition, various coexisting hidden attractors are obtained and the complex transient transition behaviors are also observed. More interestingly, the unusual and striking dynamic behavior of the coexistence of infinitely many hidden attractors is revealed by selecting the different initial values of the system, which means that extreme multistability arises. The rich and complex hidden dynamic characteristics of this system are investigated by phase portraits, bifurcation diagrams, Lyapunov exponents, and so on. Finally, the new system is implemented by an electronic circuit. A very good agreement is observed between the experimental results and the numerical simulations of the same system on the Matlab platform.
本文通过在三维混沌系统中使用简单的状态反馈控制器，推导出了一种新型的四维混沌系统。该系统状态方程由九项组成，其中仅包含一项常数项。根据常数项的不同取值，新提出的系统存在一条平衡点线或不存在平衡点。与其他类似混沌系统相比，新提出的系统具有更丰富和复杂的动力学特性。我们感兴趣的是，如果系统常数项的值非零，则它没有平衡点，因此，由于缺乏异宿轨道或同宿轨道，希尔尼科夫定理不适用于验证混沌的存在。然而，从这个新系统中可以得到单翼、双翼、三翼和四翼隐藏吸引子。此外，还得到了各种共存的隐藏吸引子，并观察到了复杂的瞬态过渡行为。更有趣的是，通过选择系统的不同初始值，揭示了无限多个隐藏吸引子共存的异常且显著的动力学行为，这意味着出现了极端多稳态。通过相图、分岔图、李雅普诺夫指数等对该系统丰富而复杂的隐藏动力学特性进行了研究。最后，通过电子电路实现了新系统。在实验结果与同一系统在Matlab平台上的数值模拟之间观察到了非常好的一致性。
