Bifurcation theory has been very successful in the study of qualitative changes in nonlinear dynamical systems. An important tool of this theory is the existence of a center manifold near nonhyperbolic equilibria and limit cycles or homoclinic orbits. The existence has already been proven for many kinds of different systems, but not fully for limit cycles in delay differential equations (DDEs). In this paper, we prove the existence of a smooth finite-dimensional periodic center manifold near a nonhyperbolic cycle in DDEs and the existence of a special coordinate system on this manifold. This allows us to describe the local dynamics on the center manifold in terms of the standard normal forms. These results are based on the rigorous functional analytic perturbation framework for dual semigroups, also called sun-star calculus.
In this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov-Takens bifurcation in classical delay differential equations (DDEs). Using a generalization of the Lindstedt-Poincaré method to approximate the homoclinic solution allows us to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas, which have been implemented in the freely available bifurcation software package DDE-BifTool. The effectiveness is demonstrated on various models.
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