Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs

TL;DR

This work extends periodic normal-form theory from finite-dimensional ODEs to classical delay differential equations by establishing periodic center manifolds near nonhyperbolic cycles and constructing a coordinate system on the center manifold via time-periodic Jordan chains. Using sun-star calculus and evolution semigroups, the authors develop dual periodic Jordan chains for the linear and adjoint systems, prove duality relations and spectral equalities among the associated unbounded operators, and derive three normal-form theorems that describe local dynamics on the center manifold in terms of periodic normal forms. The results show that, despite the infinite-dimensional setting, the critical periodic normal forms for bifurcations of limit cycles in DDEs are the same as in ODEs, and they pave the way for explicit computation of normal-form coefficients in forthcoming work. The framework also generalizes to broader classes of sun-star delay equations and suggests avenues for implementing these coefficients in public software, with potential extensions to renewal equations and infinite-delay systems.

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.

Figures (1)

• Figure 1: Illustration of two-dimensional center manifolds $\mathcal{W}_{\mathop{\mathrm{loc}}\nolimits}^{c}(\Gamma)$ together with the coordinate system $(\tau,\xi)$. The left figure represents the case when $-1 \not \in \Lambda_0$ and then $\mathcal{W}_{\mathop{\mathrm{loc}}\nolimits}^{c}(\Gamma)$ is locally diffeomorphic to a cylinder in a neighborhood of $\Gamma$, see \ref{['thm:normalformII']}. The right figure represents the case when $-1 \in \Lambda_0$ and then $\mathcal{W}_{\mathop{\mathrm{loc}}\nolimits}^{c}(\Gamma)$ is locally diffeomorphic to a Möbius band in a neighborhood of $\Gamma$, see \ref{['thm:normalformIII']}.

Theorems & Definitions (61)

• Lemma 1
• proof
• Definition 2: Clement1988
• Remark 3
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6
• proof