On local attraction properties and a stability index for heteroclinic connections
On local attraction properties and a stability index for heteroclinic connections
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关于异宿连接的局部吸引特性和稳定性指数
DOI：
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发表时间：
2010
期刊：
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通讯作者：
P. Ashwin
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文献类型：
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作者：
O. Podvigina;P. Ashwin
一些不变集可能吸引附近的一组初始条件，但仍然排斥附近互补的一组初始条件。对于给定的具有吸引盆N的不变集，我们定义了点x的稳定性指数σ(X)，它刻画了吸引盆的局部范围∊X。设Bϵ表示半径为ϵ的关于x的球，如果σ(X)>；0，则Bϵ∖N相对于球的测度的测度是O(ϵ|σ(X)|)，而如果σ(X)<；0，则Bϵ∩N相对于球的测度的测度是这样的。我们证明了这个指数沿轨道是常量，并且我们将这个轨道不变量与其他稳定性概念联系起来，例如Milnor吸引、本质渐近稳定性和相对于正测度集的渐近稳定性。我们使定义适用于局部吸引盆地(即，N被定义为盆地中的一组初始条件，其轨迹保持在X的局部)。这个稳定性指标对于讨论鲁棒异宿周期的稳定性特别有用，其中几位作者已经研究了作为Milnor吸引子的周期附近不稳定尖点的出现。我们研究了简单(鲁棒异宿)循环，证明了局部稳定性指数(从而局部稳定性性质)可以根据周期上定态矢量场的线性化的本征值来计算。在这方面，我们推广了Krupa和Melbourne(1995)的结果。理论上是这样。系统15 121-48；2004年程序R.Soc.埃丁布。A1341177-97)，给出了简单异宿圈是Milnor吸引子的判据。
Some invariant sets may attract a nearby set of initial conditions but nonetheless repel a complementary nearby set of initial conditions. For a given invariant set with a basin of attraction N, we define a stability index σ(x) of a point x ∊ X that characterizes the local extent of the basin. Let Bϵ denote a ball of radius ϵ about x. If σ(x) > 0, then the measure of Bϵ ∖ N relative the measure of the ball is O(ϵ|σ(x)|), while if σ(x) < 0, then the measure of Bϵ ∩ N relative the measure of the ball is of this order. We show that this index is constant along trajectories, and we relate this orbit invariant to other notions of stability such as Milnor attraction, essential asymptotic stability and asymptotic stability relative to a positive measure set. We adapt the definition to local basins of attraction (i.e. where N is defined as the set of initial conditions that are in the basin and whose trajectories remain local to X). This stability index is particularly useful for discussing the stability of robust heteroclinic cycles, where several authors have studied the appearance of cusps of instability near cycles that are Milnor attractors. We study simple (robust heteroclinic) cycles in and show that the local stability indices (and hence local stability properties) can be calculated in terms of the eigenvalues of the linearization of the vector field at steady states on the cycle. In doing this, we extend previous results of Krupa and Melbourne (1995 Ergod. Theory Dyn. Syst. 15 121–48; 2004 Proc. R. Soc. Edinb. A 134 1177–97) and give criteria for simple heteroclinic cycles in to be Milnor attractors.