常微分方程系统中内部激变现象的研究

应用广义胞映射图论方法研究常微分方程系统的激变.揭示了边界激变是由于混沌吸引子与 在其吸引域边界上的周期鞍碰撞产生的，在这种情况下，当系统参数通过激变临界值时，混 沌吸引子连同它的吸引域突然消失，在相空间原混沌吸引子的位置上留下了一个混沌鞍.研 究混沌吸引子大小(尺寸和形状)的突然变化，即内部激变.发现这种混沌吸引子大小的突然 变化是由于混沌吸引子与在其吸引域内部的混沌鞍碰撞产生的，这个混沌鞍是相空间非吸引 的不变集，代表内部激变后混沌吸引子新增的一部分.同时研究了这个混沌鞍的形成与演化. 给出了对永久自循环胞集和瞬态自循环胞集进行局部细化的方法. 关键词： 广义胞映射 有向图 激变 混沌鞍

FURTHER STUDY OF INTERIOR CRISES IN SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Crises in systems of ordinary differential equations are investigated by means o f Generalized Cell Mapping Digraph (GCMD) method. We show that a boundary crisis results from a collision between a chaotic attractor and a periodic saddle on i ts basin boundary. In such a case the chaotic attractor, together with its basin of attraction, is suddenly destroyed as the parameter passes through a critical value, leaving behind a nonattracting chaotic saddle in the place of the origin al chaotic attractor in phase space. We focus here on a sudden change in the siz e of a chaotic attractor, namely an interior crisis. We demonstrate that at an i nterior crisis the chaotic attractor collides with a chaotic saddle within its b asin of attraction. This chaotic saddle is an invariant and nonattracting set an d resembles the new portion of the larger chaotic attractor just after the inter ior crisis. We also investigate the origin and evolution of the chaotic saddle. The local refining procedures of persistent and transient self-cycling sets are given.