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Abstract:
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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 . |