Switching from codimension 2 bifurcations of equilibria in delay differential equations
MetadataShow full item record
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.