Traveling Waves Arising from Neimark–Sacker Bifurcations with Discrete Cyclic Symmetry

We consider Neimark–Sacker bifurcations in discrete-time dynamical systems with discrete cyclic symmetry. We prove that, when the dynamics on a unique invariant closed curve have an irrational rotation number and satisfy the hypotheses of Denjoy’s theorem, the curve admits a traveling-wave representation: its components are generated from a single periodic profile by discrete spatial shifts. Under an additional Diophantine condition, the regularity of the profile is inherited from the dynamics on the curve. This representation reduces the dimension of the conventional parametrization problem by a factor of [Formula: see text], where [Formula: see text] is the order of the cyclic symmetry. Numerical continuation of the resulting functional equation illustrates the theory and reveals secondary phenomena including double-covering bifurcations and bifurcations to higher-dimensional invariant tori. The examples also demonstrate that discrete cyclic symmetry alone does not guarantee a traveling-wave representation once the dynamics on the invariant curve becomes periodic.