Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs

TL;DR

This paper proves the existence of a smooth, locally invariant center manifold near nonhyperbolic limit cycles in ODEs using the Lyapunov-Perron method, and provides explicit examples illustrating various smoothness properties.

Contribution

It offers an elementary proof of the center manifold existence near nonhyperbolic cycles and presents explicit examples with different smoothness and analyticity properties.

Findings

Existence of periodic smooth center manifolds proved

Explicit examples of vector fields with various smoothness properties

Elementary proof using Lyapunov-Perron method

Abstract

In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov-Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)-$C^{\infty}$-smooth and (non)-analytic periodic center manifolds.