具有共存吸引子的混沌系统及其分数阶系统的镇定

Chaotic system with coexisting attractors and the stabilization of its fractional order system

研究一个具有共存吸引子的混沌系统及对应分数阶系统的镇定问题.提出了一个新的具有双翼与四翼吸引子共存的混沌系统,利用Lyapunov指数谱和分岔图对系统的性质进行了分析.借助于拓扑马蹄理论和数值计算,找到了系统的拓扑马蹄,并获得了拓扑熵.构造了相应的分数阶混沌系统,此系统亦存在两个孤立的双翼吸引子以及四翼吸引子且共存的双翼吸引子之间没有重叠.设计了线性反馈标量控制器,此控制器用于分数阶混沌系统的镇定.在控制过程中并未删除系统的非线性项,理论分析与仿真计算表明了该方法的有效性.

A chaotic system with coexisting attractors and the stabilization problem of the corresponding fractional order system are studied. A novel chaotic system with the existence of double-wing and four-wing chaotic attractors is proposed. Its math characteristics are investigated by Lyapunov exponents spectrum and bifurcation diagram. By means of topological horseshoe theory and numerical computation, the topological horseshoe and the topological entropy in the system are obtained. Based on the system, a new 3D fractional order chaotic system is constructed. The fractional order system also has two isolated double-wing attractors and four-wing attractors, and there are not overlaps between the coexisting double wing attractors. For the stabilization of the fractional order system, a linear feedback scalar controller is designed. The nonlinear terms in the system are not deleted by the controlling method. The theoretical analysis and numerical simulation show the effectiveness of the method.