On Modal Properties of the Koopman Operator for Nonlinear Systems with Symmetry

This paper explores how symmetry in nonlinear dynamical systems is reflected in the modal properties of the corresponding Koopman operator. We first observe that symmetry in the dynamics is reflected in the symmetry of the Koopman eigenfunctions, as well as presence of repeated Koopman eigenvalues. Next, we show that the resulting infinite-dimensional linear system (constructed using independent Koopman eigenfunctions) also has an inherent symmetry. The proposed results are supported by an analytical example.