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Bifurcations of basins of attraction from the view point of prime ends

机译：从质点的角度看吸引盆的分叉

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In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is nonempty and the basins often have fractal basin boundaries. The purpose of this paper is to describe the structure and properties of unbounded basins and their boundaries for two-dimensional diffeomorphisms. Frequently, if not always, there is a periodic saddle on the boundary that is accessible from the basin. Caratheodory and many others developed an approach in which an open set (in our case a basin) is compactified using so-called prime end theory. Under the prime end compactification of the basin, boundary points of the basin (prime ends) can be characterized as either type 1, 2, 3, or 4. In all well-known examples, most points are of type 1. Many two-dimensional basins have a basin cell, that is, a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well chosen periodic orbit. Then the basin consists of a central body (the basin cell) and a finite number of channels attached to it, and the basin boundary is fractal. We present a result that says {a basin has a basin cell} if and only if {every prime end that is defined by a chain of unbounded regions (in the basin) is a prime end of type 3 and furthermore all other prime ends are of type 1}. We also prove as a parameter is varied, the basin cell for a basin B is created (or destroyed) if and only if either there is a saddle node bifurcation or the basin B has a prime end that is defined by a chain of unbounded regions and is a prime end of either type 2 or type 4.
机译：在动力学系统中，两个或多个吸引子共存的例子很常见，在这种情况下，盆地边界是非空的，盆地经常具有分形盆地边界。本文的目的是描述二维无定形的无界盆地的结构和性质及其边界。通常（如果不是总是这样），在盆地的边界上有一个周期性的鞍状区域，可以从盆地中进入。 Caratheodory和其他许多人开发了一种方法，其中使用所谓的素端理论将开放集（在我们的情况下为盆地）压实。在盆地的原始端压实下，盆地的边界点（初始端）可以被定义为类型1、2、3或4。在所有众所周知的示例中，大多数点都是类型1。维盆地具有盆地单元，即一个圈闭区，其边界由精心选择的周期性轨道的稳定和不稳定歧管组成。然后，盆地由中心主体（盆地单元）和与其相连的有限数量的通道组成，盆地边界是分形的。当且仅当{由无边界区域链定义的每个主要端点（在盆地中）定义为类型3的主要端点，并且所有其他主要端点为{类型1}。我们还证明，随着参数的变化，当且仅当存在鞍形节点分叉或盆地B的初始端由无边界区域链定义时，才会创建（或破坏）盆地B的盆地单元并且是类型2或类型4的质数端。

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