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Numerical Periodic Normalization at Codim 1 Bifurcations of Limit Cycles in DDEs

M. M. Bosschaert, B. Lentjes, L. Spek, Yu. A. Kuznetsov

TL;DR

This work derives explicit, computable formulas for the critical normal form coefficients of codimension-one bifurcations of limit cycles in delay differential equations by combining periodic center manifolds with sun-star calculus and a novel characteristic operator Δ(z). The approach yields compact expressions for fold, period-doubling, and Neimark-Sacker bifurcation coefficients, enabling robust numerical evaluation via orthogonal collocation on periodic BVPs and bordered inverses, implemented in the Julia package PeriodicNormalizationDDEs. It avoids Poincaré maps, works directly with the center manifold reductions, and provides a unifying framework that mirrors finite-dimensional ODE results while addressing periodic spectral challenges through the adjoint characterstic operator. Applications to neural-field and active-control models demonstrate correctness, efficiency gains over pseudospectral methods, and practical utility for bifurcation analysis of limit cycles in DDEs. The methodology paves the way for codimension-two extensions and broader applicability to periodic evolution equations beyond classical DDEs.

Abstract

Recent work in [53, 54] by the authors on periodic center manifolds and normal forms for bifurcations of limit cycles in delay differential equations (DDEs) motivates the derivation of explicit computational formulas for the critical normal form coefficients of all codimension one bifurcations of limit cycles. In this paper, we derive such formulas via an application of the periodic normalization method in combination with the functional analytic perturbation framework for dual semigroups (sun-star calculus). The explicit formulas allow us to distinguish between nondegenerate, sub- and supercritical bifurcations. To efficiently apply these formulas, we introduce the characteristic operator as this enables us to use robust numerical boundary-value algorithms based on orthogonal collocation. Although our theoretical results are proven in a more general setting, the software implementation and examples focus on discrete DDEs. The actual implementation is described in detail and its effectiveness is demonstrated on various models.

Numerical Periodic Normalization at Codim 1 Bifurcations of Limit Cycles in DDEs

TL;DR

This work derives explicit, computable formulas for the critical normal form coefficients of codimension-one bifurcations of limit cycles in delay differential equations by combining periodic center manifolds with sun-star calculus and a novel characteristic operator Δ(z). The approach yields compact expressions for fold, period-doubling, and Neimark-Sacker bifurcation coefficients, enabling robust numerical evaluation via orthogonal collocation on periodic BVPs and bordered inverses, implemented in the Julia package PeriodicNormalizationDDEs. It avoids Poincaré maps, works directly with the center manifold reductions, and provides a unifying framework that mirrors finite-dimensional ODE results while addressing periodic spectral challenges through the adjoint characterstic operator. Applications to neural-field and active-control models demonstrate correctness, efficiency gains over pseudospectral methods, and practical utility for bifurcation analysis of limit cycles in DDEs. The methodology paves the way for codimension-two extensions and broader applicability to periodic evolution equations beyond classical DDEs.

Abstract

Recent work in [53, 54] by the authors on periodic center manifolds and normal forms for bifurcations of limit cycles in delay differential equations (DDEs) motivates the derivation of explicit computational formulas for the critical normal form coefficients of all codimension one bifurcations of limit cycles. In this paper, we derive such formulas via an application of the periodic normalization method in combination with the functional analytic perturbation framework for dual semigroups (sun-star calculus). The explicit formulas allow us to distinguish between nondegenerate, sub- and supercritical bifurcations. To efficiently apply these formulas, we introduce the characteristic operator as this enables us to use robust numerical boundary-value algorithms based on orthogonal collocation. Although our theoretical results are proven in a more general setting, the software implementation and examples focus on discrete DDEs. The actual implementation is described in detail and its effectiveness is demonstrated on various models.

Paper Structure

This paper contains 30 sections, 15 theorems, 168 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background, aims and challenges
  3. Overview
  4. Periodic center manifolds and normal forms for DDEs
  5. Periodic normal forms for codim 1 bifurcations
  6. Fold bifurcation
  7. Period-doubling bifurcation
  8. Neimark-Sacker bifurcation
  9. Periodic spectral computations
  10. Periodic smooth Jordan chains via characteristic operators
  11. Solvability of periodic linear operator equations
  12. Periodic normalization on the center manifold
  13. Computation of critical coefficients
  14. Fold bifurcation
  15. Period-doubling bifurcation
  16. ...and 15 more sections

Key Result

Lemma 4

The operator $\Delta(z)$ is closed and $\rho(\Delta(z)) \neq \emptyset$ for all $z \in \mathbb{C}$, and there holds

Figures (2)

  • Figure 1: Bifurcation diagrams of \ref{['eq:neocortex']} with unfolding parameters $(a,c)$. Left, sub- and supercritical period-doubling (PD) branches are shown, with two identified generalized period-doubling points connected by a limit point of cycles branch. Right, a close-up near the cusp bifurcation located on the LPC-branch.
  • Figure 2: Bifurcation diagrams of \ref{['eq:activecontrol']} with unfolding parameters $(\zeta,\tau)$. The left figure is obtained via periodic normalization in DDEs while right figure is obtained via a pseudospectral approximation.

Theorems & Definitions (35)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 25 more