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dc.rights.licenseCC-BY-NC-ND
dc.contributor.advisorKuznetsov, Prof. Dr. Yuri A.
dc.contributor.authorBosschaert, M.M.
dc.date.accessioned2016-07-14T17:00:43Z
dc.date.available2016-07-14T17:00:43Z
dc.date.issued2016
dc.identifier.urihttps://studenttheses.uu.nl/handle/20.500.12932/22663
dc.description.abstractSmooth 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.
dc.description.sponsorshipUtrecht University
dc.format.extent3375750
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.titleSwitching from codimension 2 bifurcations of equilibria in delay differential equations
dc.type.contentMaster Thesis
dc.rights.accessrightsOpen Access
dc.subject.keywordsdelay 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
dc.subject.courseuuMathematical Sciences


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