Wada分形吸引域边界上的混沌鞍

A CHAOTIC SADDLE IN A WADA FRACTAL BASIN BOUNDARY

应用广义胞映射图论（Ｇｅｎｅｒａｌｉｚｅｄ Ｃｅｌｌ Ｍａｐｐｉｎｇ Ｄｉｇｒａｐｈ）方法，数值地研究Ｔｈｏｍｐｓｏｎ的逃逸方程在最佳逃逸点附近的分岔．发现了嵌入在Ｗａｄａ分形吸引域边界上的混沌鞍，混沌鞍是状态空间不稳定（非吸引）的混沌不变集合，Ｗａｄａ分形吸引域边界是具有Ｗａｄａ性质的边界，即吸引域边界上的任意点也同时是至少两个其它吸引域的边界点，称为Ｗａｄａ 域边界．我们证明Ｗａｄａ 域边界上的混沌鞍导致局部鞍结分岔具有全局不确定性结局，研究了Ｗａｄａ域边界上混沌鞍的形成与演化，证明最终的逃逸分岔是混沌吸引子碰撞混沌鞍的边界激变．

In this paper, bifurcations near optimal escape for Thompson's escape equation are numerically studied by means of Generalized Cell Mapping Digraph (GCMD) method. We find a chaotic saddle embedded in a Wada fractal basin boundary. The chaotic saddle is an unstable (nonattracting) chaotic invariant set. The Wada fractal basin boundary has the Wada property that any point that is on the boundary of that basin is also simultaneously on the boundary of at least two other basins. The chaotic saddle in the Wada basin boundary plays an extremely important role in the bifurcations governing the escape. We demonstrate that the chaotic saddle in the Wada basin boundary leads to a local saddle-node fold bifurcation with globally indeterminate outcome. In such a case, the attractor (node) and the saddle of the saddle-node fold are merged into the chaotic saddle and the chaotic saddle also undergoes an abrupt enlargement in its size as a parameter passes through the bifurcation value, simultaneously the Wada basin boundary is also converted into the fractal basin boundary of two remaining attractors, in particular, the chaotic saddle after the saddle-node fold bifurcation is in the fractal basin boundary, this implies that the saddle-node fold bifurcation has indeterminate outcome, namely, after the system drifts through the bifurcation, which of the two remaining attractors the orbit goes to is indeterminate in that it is sensitively dependent on arbitrarily small effects such as how the parameter is changed and/or noise and/or computer roundoff, obviously, this presents an extreme form of indeterminacy in a dynamical system. We also investigate the origin and evolution of the chaotic saddle in the Wada basin boundary and demonstrate that the chaotic saddle in the Wada basin boundary is created by the collision between two chaotic saddles in different fractal basin boundaries. We demonstrate that a final escape bifurcation is the boundary crisis caused by the collision between a chaotic attractor and a chaotic saddle, and this implies that Grebogi's definition of the boundary crisis by the collision with a periodic saddle is generalized.