Applied Symbolic Dynamics

TL;DR

This paper reviews recent advances in applied symbolic dynamics, emphasizing its utility in analyzing complex nonlinear systems through coarse-grained, geometry-aware symbolic representations, including for higher-dimensional maps and differential equations.

Contribution

It provides a comprehensive overview of recent developments in applying symbolic dynamics to complex systems, extending its use to two-dimensional maps and differential equations.

Findings

Symbolic dynamics can be developed for multi-dimensional mappings.

It enables analysis of ordinary differential equations via Poincaré maps.

Recent advances enhance practical applications in nonlinear science.

Abstract

Symbolic dynamics is a coarse-grained description of dynamics. By taking into account the ``geometry'' of the dynamics, it can be cast into a powerful tool for practitioners in nonlinear science. Detailed symbolic dynamics can be developed not only for one-dimensional mappings, unimodal as well as those with multiple critical points and discontinuities, but also for some two-dimensional mappings. The latter paves the way for symbolic dynamics study of ordinary differential equations via the Poincar\'e maps. This paper provides an overview of the recent development of the applied aspects of symbolic dynamics.