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Center Manifolds and Normal Forms for Nonlinearly Periodically Forced DDEs

Bram Lentjes, Seppe Daniëls, Meinder Follon, Yuri A. Kuznetsov

TL;DR

The paper addresses bifurcations of equilibria in nonlinearly periodically forced delay differential equations by developing a periodic center-manifold theorem within the sun-star framework, and by deriving a periodic normal-form theory with explicit coefficient formulas for the periodically forced fold and nonresonant Hopf bifurcations. Solutions near equilibria on the center manifold are reduced to finite-dimensional, periodically forced ODEs, enabling straightforward computation of normal-form coefficients. The authors provide rigorous constructions (center manifolds, parametrizations) and practical normalization formulas, illustrated by examples like a periodic fold model and the Wright equation, demonstrating how periodic forcing reshapes bifurcation diagrams and resonance phenomena. The results offer a concrete, implementable toolkit for analyzing nonautonomous DDEs and pave the way for software integration and parameter-dependent extensions with broad applicability. Overall, the work significantly advances bifurcation analysis for nonautonomous delay systems by unifying center-manifold theory, periodic normal forms, and explicit coefficient calculations within a rigorous functional-analytic framework.

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.

Center Manifolds and Normal Forms for Nonlinearly Periodically Forced DDEs

TL;DR

The paper addresses bifurcations of equilibria in nonlinearly periodically forced delay differential equations by developing a periodic center-manifold theorem within the sun-star framework, and by deriving a periodic normal-form theory with explicit coefficient formulas for the periodically forced fold and nonresonant Hopf bifurcations. Solutions near equilibria on the center manifold are reduced to finite-dimensional, periodically forced ODEs, enabling straightforward computation of normal-form coefficients. The authors provide rigorous constructions (center manifolds, parametrizations) and practical normalization formulas, illustrated by examples like a periodic fold model and the Wright equation, demonstrating how periodic forcing reshapes bifurcation diagrams and resonance phenomena. The results offer a concrete, implementable toolkit for analyzing nonautonomous DDEs and pave the way for software integration and parameter-dependent extensions with broad applicability. Overall, the work significantly advances bifurcation analysis for nonautonomous delay systems by unifying center-manifold theory, periodic normal forms, and explicit coefficient calculations within a rigorous functional-analytic framework.

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.
Paper Structure (18 sections, 18 theorems, 143 equations, 5 figures)

This paper contains 18 sections, 18 theorems, 143 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Challenges and overview
  3. Periodically forced center manifold theorem
  4. Functional analytic framework
  5. Spectral decomposition and the linear (in)homogeneous equation
  6. Modification of the nonlinearity
  7. Existence of a Lipschitz center manifold
  8. Properties of the center manifold
  9. The special case of classical DDEs
  10. Periodically forced normal form theorem
  11. Spectral computations for periodically forced DDEs
  12. Normalization for periodically forced DDEs
  13. Periodically forced fold bifurcation
  14. Periodically forced nonresonant Hopf bifurcation
  15. Examples
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose that $T$ is eventually norm continuous and let $\sigma(A)$ be the pairwise disjoint union of where $\sigma_{-}$ is closed, while $\sigma_0$ and $\sigma_{+}$ are compact. If $\sup_{\lambda \in \sigma_{-}} \Re \lambda < 0 < \inf_{\lambda \in \sigma_{+}} \Re \lambda$, then hyp:1 and hyp:2 hold for $T$ on $X$.

Figures (5)

  • Figure 1: Vector field of the periodically forced ODE \ref{['eq:foldPFexample']}, shown together with the invariant sets $\Gamma_{\pm}$ and $\mathbb{R} \times \{\overline{x}\}$ for various values of the unfolding parameter $\beta$, while keeping $b = 1$ and $N(t,\xi(t)) = \sin(t)$ fixed.
  • Figure 2: Forward orbits $\mathcal{O}_\beta$ of the periodically forced ODE \ref{['eq:HopfPFexample']}, shown together with the invariant sets $C_\beta$ and $\mathbb{R} \times \{\overline{x}\}$ for various values of the unfolding parameter $\beta$, while keeping $\omega = \pi, c=-1$ and $N(t,\xi(t)) = \sin(t)$ fixed.
  • Figure 3: Illustration of a nondegenerate periodically forced fold bifurcation in the truncated scalar DDE \ref{['eq:examplefoldalpha2']} with $\beta_1 = 1$ and $\beta_2(t) = 1 + \sin(t)$. Shown are the invariant sets $\Gamma_{\pm}$ ($2\pi$-periodic hyperbolic limit cycles) and $\mathbb{R} \times \{0\}$ (nonhyperbolic equilibrium) together with several forward orbits $\mathcal{O}(\alpha_1)$ that are computed with initial conditions $\varphi \in \{0,\pm\frac{1}{4}, \pm \frac{1}{2}, \pm \frac{3}{4}, \pm 1\} \subseteq X$.
  • Figure 4: Bifurcation diagram of the nonlinearly periodically forced Wright equation \ref{['eq:wright']} with forcing function $\beta(t) = \cos(\Omega_2t)$ and unfolding parameters $(\Omega_2,a)$. Horizontal lines with $a \neq 0$ correspond to different Hopf branches.
  • Figure 5: Several forward orbits $\mathcal{O}(\Omega_2,a)$ computed using the initial condition $\varphi=1 \in X$, together with the invariant sets $\mathbb{R} \times \{\overline{x}\}$ and $C(\Omega_2,a)$, illustrated for various parameter values $(\Omega_2,a)$ of the periodically forced Wright equation \ref{['eq:wright']} with forcing function $\beta(t)=\cos(\Omega_2 t)$, shown in $(t,x(t),x(t-1))$ space.

Theorems & Definitions (37)

  • Theorem 1: Janssens2020
  • Lemma 2
  • Proposition 3
  • proof
  • Theorem 4
  • proof
  • Proposition 5
  • Proposition 6
  • Theorem 7: Local center manifold
  • Remark 8
  • ...and 27 more