The structure of basins of attraction and their trapping regions.

*(English)*Zbl 0893.58039The purpose of the paper is to describe the structure and properties of basins of attraction and their boundaries for two-dimensional diffeomorphisms. Diffeomorphisms of a two-dimensional smooth boundaryless manifold \(M\) are assumed to have at least three basins of attraction. The main questions addressed in the paper are the following: When there are three basins of attraction, is it possible that every boundary point of one basin is on the boundary of the remaining basins? Is it possible that all three boundaries coincide?

To give answers to these and some other questions, the authors introduce the basic notion of a “basin cell” that plays a fundamental role in their investigations. Roughly speaking, a basin cell is a trapping region generated by some well chosen saddle-hyperbolic periodic orbit and determines the structure of the corresponding basin. Several characteristic properties of basin cells are established.

Basin cells are primarily used to state conditions ensuring the “Wada property” of basins. A basin \(B\) is called a Wada basin if every \(x\in \partial \overline B\) is a Wada point, i.e., if every open neighborhood of \(x\) in \(M\) has a nonempty intersection with at least three different basins. Assuming that \(B\) is the basin of a basin cell (generated by a periodic orbit \(P\)), the authors show that \(B\) is a Wada basin if the unstable manifold of \(P\) intersects at least three basins. This result implies conditions for basins \(B_1, B_2,\dots ,B_N\) (\(N>3\)) to satisfy \(\partial\overline B_1=\partial\overline B_2=\cdots =\partial\overline B_N\). Results obtained in the paper provide numerically verifiable conditions guaranteeing the “Wada property” of a dynamical system. Several specific dynamical systems (the Hénon map and the forced damped pendulum equation) are considered.

To give answers to these and some other questions, the authors introduce the basic notion of a “basin cell” that plays a fundamental role in their investigations. Roughly speaking, a basin cell is a trapping region generated by some well chosen saddle-hyperbolic periodic orbit and determines the structure of the corresponding basin. Several characteristic properties of basin cells are established.

Basin cells are primarily used to state conditions ensuring the “Wada property” of basins. A basin \(B\) is called a Wada basin if every \(x\in \partial \overline B\) is a Wada point, i.e., if every open neighborhood of \(x\) in \(M\) has a nonempty intersection with at least three different basins. Assuming that \(B\) is the basin of a basin cell (generated by a periodic orbit \(P\)), the authors show that \(B\) is a Wada basin if the unstable manifold of \(P\) intersects at least three basins. This result implies conditions for basins \(B_1, B_2,\dots ,B_N\) (\(N>3\)) to satisfy \(\partial\overline B_1=\partial\overline B_2=\cdots =\partial\overline B_N\). Results obtained in the paper provide numerically verifiable conditions guaranteeing the “Wada property” of a dynamical system. Several specific dynamical systems (the Hénon map and the forced damped pendulum equation) are considered.

Reviewer: E.Ershov (St.Peterburg)

##### MSC:

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |