The symbolic partition with generalized Koopman analysis

Abstract

Symbolic dynamics serves the study of chaotic systems as a crucial tool, prompting extensive research on a proper partition of the phase space. However, the majority of prevailing methods are empirical and based either on construction of manifolds or on specific orbits. A spectral consideration based on evolution operators, such as the Koopman operator, remains underexplored. Here, we propose an alternative method for the symbolic partition based on the Koopman operator, the left eigenfunctions of which turn out closely related to the stretching and folding mechanism of chaos generation and thus provide novel means to identify a partition boundary. To avoid wild oscillations in eigenfunctions, a generalized Koopman analysis is developed to enable a local spectral computation for refining a partition. This new framework is successfully demonstrated in several typical dynamical systems including 1-D chaotic maps, the well-known Hénon maps with different parameters and a 3-D hyperchaotic map as well as a periodically driven Duffing system, with or without small noisy perturbation. Thus, the proposed technique is highly flexible and good for chaotic systems in multi-dimensions with diverse complexities.