Classification of symmetric chaos using the Perron–Frobenius and Koopman operators

Abstract

The symmetry properties of the attractors of equivariant dissipative dynamical systems can suffer symmetry-changing bifurcations, that can be detected and classified using well-established methods. Novel data-driven methods, such as the Koopman and Perron–Frobenius operators, besides reducing any nonlinear system to a linear one, can also be applied to the analysis of equivariant dynamical systems and the classification problem. In this article, we study matrix approximations of these infinite-dimensional operators that respect the original symmetry and introduce an aggregate matrix that has a clear interpretation. Its sparsity pattern reveals the presence of multiple conjugate attractors and indicates the structure of their symmetry subgroup. We apply these ideas to data generated by three nonlinear equivariant systems, finding attractors of non-trivial subgroups and detecting symmetry-changing bifurcations. The proposed method can be incorporated into existing computational processes for the analysis, prediction and control of nonlinear equivariant systems.