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Switching from codimension 2 bifurcations of equilibria in delay differential equations

by MM Bosschaert

Institution: Universiteit Utrecht
Year: 2016
Keywords: delay differential equations; sun-star calculus; parameter-dependent normal forms; numerical bifurcation analysis; DDE-BifTool; predictors; (transcritical) Bogdanov-Takens; generalized Hopf; zero-Hopf; Hopf-transcritical; Hopf-Hopf
Posted: 02/05/2017
Record ID: 2135162
Full text PDF: http://dspace.library.uu.nl:8080/handle/1874/334792


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. Advisors/Committee Members: Kuznetsov, Prof. Dr. Yuri A..

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