分数阶统一混沌闪烁系统中的幽灵吸引子

Ghost attractors in fractional order unified chaotic blinking systems

中文关键词  分数阶微积分  闪烁系统  混沌  Hopf分岔

英文关键词  fractional calculus  blinking system  chaos  Hopf bifurcation

中文摘要

      目前分数阶随机闪烁系统的动力学研究甚少, 本文考虑具有拓扑不等价子系统耦合的分数阶统一混沌闪 烁系统. 首先, 本文证明了分数阶统一混沌系统的平衡点的局部稳定性和Hopf分岔条件; 其次, 本文分析了随机周 期增加和分数阶算子的阶数增加时动力学行为的变化. 结果表明, 在特定的切换概率下, 分数阶统一混沌闪烁系统 存在幽灵吸引子. 在快速切换情况下, 随着切换周期的增加, Lorenz吸引子和Chen吸引子之间切换所产生的吸引子 会迅速偏离作为幽灵混沌吸引子的Lu¨吸引子; 而在非快速切换情况下, 随着切换周期的增加, 非稳态吸引子与幽灵 吸引子之间的偏离程度将保持稳定; 同时, 这两个吸引子的偏离程度会依赖于分数阶算子的阶数α. 基于遍历性质, 本文对闪烁系统和其平均系统的不变测度进行了数值分析, 估计了非稳态吸引子和幽灵吸引子的接近程度.

英文摘要

      Little seems to be known about the dynamics of the fractional order stochastically blinking systems. In this paper, we consider the complex dynamics of fractional order unified chaotic blinking systems coupled topologically nonequivalent subsystems. Firstly, we prove the conditions for local stability and Hopf bifurcation of the fractional order unified chaotic system. Then we analyze the change in the dynamical behavior with the increasing of the period of stochastic switching and fractional order. We demonstrate that under special switching probabilities, the fractional order unified chaos blinking system exhibits ghost attractor. Fast switching between a Lorenz attractor and a Chen attractor yields an attractor which may rapidly deviate from the Lu attractor which acts as a ghost chaotic attractor as the switching period increases. ¨ Under the case of non fast switching, the degree of deviation between the non-stationary attractor and the ghost attractor will remain stable with increasing switching period. Additionally, the deviation degree of these two attractors depends on the order of the fractional operator. Based on ergodic property, we carry out the numerical approximations of the invariant measures of the blinking and averaged systems, resulting in the estimate of a non-stationary and ghost attractors’ proximity.