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Investigation of Dynamical Systems Using Symbolic Images: Efficient Implementation and Applications

Symbolic images represent a unified framework to apply several methods for the investigation of dynamical systems both discrete and continuous in time. By transforming the system flow into a graph, they allow it to formulate investigation methods as graph algorithms. Several kinds of stable and unstable return trajectories can be localized on this graph as well as attractors, their basins and connecting orbits. Extensions of the framework allow, e.g. the calculation of the Morse spectrum and verification of hyperbolicity. In this work, efficient algorithms and adequate data structures will be presented for the construction of symbolic images and some basic operations on them, like the localization of the chain recurrent set and periodic orbits. The performance of these algorithms will be analyzed and we show their application in practice. The focus is not only put on several standard systems, like Lorenz and Ikeda, but also on some less well-known ones. Additionally, some tuning techniques are presented for an efficient usage of the method.