Switching to nonhyperbolic cycles from codim-2 bifurcations of equilibria in DDEs

Abstract

Using the framework of dual semigroups, the existence of a finite dimensional smooth center manifold for DDEs can be rigorously established [1]. This makes it is possible to apply the normalization method for local bifurcations of ODEs [2] to DDEs. Recently, the critical normal form coefficients for all five codimension 2 bifurcation of equilibria in generic DDEs have been derived [7] and implemented into the Octave/Matlab package DDE-BifTool [5]. We generalize a center manifold theorem from [1] to generic parameter-dependent DDEs, covering the cases where the critical equilibrium can disappear. It allows us to initialize the continuation of codimension 1 equilibrium and nonhyperbolic cycle bifurcations emanating from the generalized Hopf, zero-Hopf and Hopf-Hopf bifurcations in DDEs, which are the only codim 2 eqillibrium bifurcations in generic DDEs where nonhyperbolic cycles could originate. The obtained expressions have been implemented in DDE-BifTool and tested on various models.