Switching from codimension 2 bifurcations of equilibria in delay differential equations

Abstract

Smooth ordinary Delay Differential Equations (DDEs) appear in many applications, including neuroscience, ecology, and engineering. The theory of local bifurcations in one-parameter families of such DDEs is well developed starting from the 1970s, while efficient methods to analyze such bifurcations in two-parameter families have only been recently understood. In particular, efficient methods to compute coefficients of the critical normal forms have been developed and implemented in the standard Matlab software DDE-BifTool for the five well-known codim 2 bifurcations of equilibria.

However, no parameter-dependent normal form reduction has been attempted, while such reduction is crucial for deriving asymptotics of codim 1 non-equilibrium solutions (e.g. saddle homoclinic orbits and non-hyperbolic cycles) emanating from some codim 2 local bifurcations.

In this thesis, a generalization of the parameter-dependent center manifold Theorem for DDEs is given. This allows us to perform the parameter-dependent center manifold reduction and normalization near generic and transcritical Bogdanov-Takens, generalized Hopf, fold-Hopf, Hopf-transcritical and Hopf-Hopf bifurcations in DDEs.

With this combined reduction-normalization technique we are now able to start the automatic continuation of homoclinic orbits near the generic and transcritical Bogdanov-Takens bifurcations, and codim 1 cycle bifurcations emanating from generalized Hopf, fold-Hopf, Hopf-transcritical and Hopf-Hopf bifurcations.

Demonstrations of the efficiency of the developed and implemented predictors on many know DDE models (a delayed feedback financial model, a neural mass model, Holling-Tanner delayed predator-prey model, two neural network models, an approximation of a DDE with state-dependent delays, and Van de Pol oscillator with various delay types) are given.