Coexistence of attractors and Wada basin boundaries in dynamical systems : a survey of results

This is a summary report on some existing results and methods regarding the problem of determining the basins of attraction of dynamical systems (in particular, two-dimensional diffeomorphisms) when there is a coexistence of attractors. Based on the work of Helena Nusse and James Yorke, it presents existence and characterization results for a certain kind of basin boundaries (namely, the Wada boundaries). The key feature of their approach is to redefine the idea of a basin boundary by introducing the notion of a `basin cell', which bypasses the problem of exactly locating the attractor of a system, which is often either not well-defined or hard to locate in practice. Moreover, the basin cells and their boundaries are characterized by utilizing the stable and unstable manifolds of the system, which are easier to locate by numerical methods, and thus their method provides both numerically verifiable characteristics and algorithms for computation.