Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

TL;DR

This paper develops a method for analyzing bifurcations in delay differential equations, enabling the continuation of cycles from codimension two bifurcations using explicit formulas implemented in DDE-BifTool.

Contribution

It derives explicit normal form coefficients for key bifurcations in DDEs and implements them in a software package for practical bifurcation analysis.

Findings

Effective continuation of cycles from bifurcation points

Explicit formulas for normal form coefficients

Demonstrated on various delay differential equation models

Abstract

In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models.