The structure of basins of attraction and their trapping regions

Abstract

In dynamical systems examples are common in which two or more attractors coexist, and in such cases, the basin boundary is nonempty. When there are three basins of attraction, is it possible that every boundary point of one basin is on the boundary of the two remaining basins? Is it possible that all three boundaries of these basins coincide? When this last situation occurs the boundaries have a complicated structure. This phenomenon does occur naturally in simple dynamical systems. The purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. We introduce the basic notion of a ‘basin cell’. A basin cell is a trapping region generated by some well chosen periodic orbit and determines the structure of the corresponding basin. This new notion will play a fundamental role in our main results. We consider diffeomorphisms of a two-dimensional smooth manifold $M$ without boundary, which has at least three basins. A point $x\in M$ is a Wada point if every open neighborhood of $x$ has a nonempty intersection with at least three different basins. We call a basin $B$ a Wada basin if every $x\in\partial\bar{B}$ is a Wada point. Assuming $B$ is the basin of a basin cell (generated by a periodic orbit $P$), we show that $B$ is a Wada basin if the unstable manifold of $P$ intersects at least three basins. This result implies conditions for basins $B_{1},B_{2},\ldots,B_{N}(N\ge 3)$ to satisfy $\partial\bar{B}_{1}=\partial\bar{B}_{2}=\cdots =\partial\bar{B}_{N}$.