高次logistic映射的混沌运动

Chaos of High Degree Logistic-Like Map

logistic映射具有简单的形式,但却有复杂的动力学性质。构造了一种高次logistic映射,对这类映射的动力学特性做了研究。通过稳定性分析,得到了定点的稳定条件。数值迭代发现,此类映射都具有相似的稳定定点经倍周期分岔进入混沌的道路,反映了典型的Feigenbaum途径。当参数超过某个临界值时存在混沌吸引子的突然消失和突然膨胀,分别称为边界激变和内部激变。特别的当参数取一定值时会出现一种内部激变,使得混沌吸引子突然增大并且关于x轴对称,这是logistic映射所没有的。分析高次logistic映射中的这种激变对动力学的研究具有一定的意义。

Logistic map is formally simple, but is proved with complicated dynamic characteristic. This article built a high degree logistic - like map, and investigated the dynamic characteristic of this kind of map. Stability conditions of the fixed points were calculated by the analysis of stability. By means of iteration, it is found that this kind of map has the similar path from stable fixed point through period doubling bifurcation to chaos. This path is a typical symbol of Feigenbaum route. The size of chaotic attractor may suddenly disappear or expand while the parameter passes through a critical value. These two phenomena are called boundary crisis and interior crisis. Especially, when a parameter is chosen, the chaotic attractor could expand suddenly and become asymmetrical about x - axis, and this interior crisis is never found in logistic map. It is valuable for the research work of dynamics to analyses the crisis phenomena in high degree logistic - like map.