Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs

TL;DR

This paper develops a framework for analyzing bifurcations of limit cycles in delay differential equations by constructing periodic normal forms and coordinate systems on the center manifold, facilitating the study of local dynamics.

Contribution

It introduces the existence of a special coordinate system on the center manifold for DDEs, enabling the description of local bifurcation dynamics via periodic normal forms.

Findings

Existence of a coordinate system on the center manifold for DDEs.

Construction of periodic normal forms near nonhyperbolic cycles.

Development of tools for analyzing local dynamics in delay differential equations.

Abstract

A recent work by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates the derivation of periodic normal forms. In this paper, we prove the existence of a special coordinate system on the center manifold that will allow us to describe the local dynamics on the center manifold near the cycle in terms of these periodic normal forms. To construct the linear part of this coordinate system, we prove the existence of time periodic smooth Jordan chains for the original and adjoint system. Moreover, we establish duality and spectral relations between both systems by using tools from the theory of delay equations and Volterra integral equations, dual perturbation theory, duality theory and evolution semigroups.