Center Manifolds and Normal Forms for Nonlinearly Periodically Forced DDEs

TL;DR

This paper develops a rigorous framework using center manifolds and normal forms to analyze bifurcations in nonlinearly periodically forced delay differential equations, enabling explicit computation of bifurcation coefficients.

Contribution

It introduces a novel functional analytic approach for constructing center manifolds and normal forms in forced DDEs, with explicit formulas for key bifurcations.

Findings

Validated the approach with examples from literature.

Derived explicit formulas for fold and Hopf bifurcation coefficients.

Confirmed the effectiveness of the framework for analyzing bifurcations.

Abstract

The aim of this paper is to provide an effective framework for analysing bifurcations of equilibria in nonlinearly periodically forced delay differential equations. First, we establish the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic equilibrium using the rigorous functional analytic framework of dual semigroups (sun-star calculus). Second, we construct a center manifold parametrization that allows us to describe the local dynamics on the center manifold near the equilibrium in terms of periodically forced normal forms. Third, we present a normalization method to derive explicit computational formulas for the critical normal form coefficients at a bifurcation of interest. In particular, we obtain such formulas for the periodically forced fold and nonresonant Hopf bifurcation. Several examples and indications from the literature confirm the validity and effectiveness of our approach.