Homoclinic solutions in finite and infinite dimensional systems

Abstract

This thesis is concerned with higher-order asymptotics to the homoclinic orbit near the generic and transcritical codimension two Bogdanov-Takens bifurcation in infinite dimensional systems generated by delay differential equations (DDEs). First, we will obtain accurate homoclinic asymptotics in the normal form. Then we will perform the parameter-dependent center manifold reduction near the generic and transcritical Bogdanov-Takens points. To achieve this, we rigorously derive a method to translate asymptotics of solutions in the normal form for a local bifurcation, to asymptotics of solutions on the parameter-dependent center manifold. In particular, we allow for a time-reparametrization in the ho-mological equation, enabling us to consider orbital normal forms. The use of orbital normal forms turns out to be particularly useful when obtaining third-order homoclinic asymptotics near the transcritical Bogdanov-Takens bifurcations. Indeed, we show that, by using orbital normal forms, these asymptotics can be obtained through a simple transformation from the generic case. Additionally, a detailed comparison is provided between applying different Bogdanov-Takens normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt-Poincaré) to approximate the homoclinic solution near Bogdanov-Takens points. In particular, we will show that higher-order time approximations to the nonlinear time transformation in the Lindstedt-Poincaré method are essential. Next to the codimension two Bogdanov-Takens bifurcation points, we also perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurca-tions emanating from these codimension two bifurcation points. Furthermore, the known existence theorem of a smooth finite-dimensional parameter-dependent center manifold for delay differential equations is generalized to allow for the equilibrium under consideration to vanish, as is the case in the zero-Hopf and the generic Bogdanov-Takens bifurcation points. The proof is given at the abstract semigroup level using the framework of perturbation theory for dual semigroups. The non-hyperbolic cycles and homoclinic asymptotics are implemented in DDE-BifTool to start numerical continuation of these homoclinic curves. The homoclinic predictor in MatCont has been corrected as well. The effectiveness of the new predictors is demonstrated in numerous examples. In-depth treatments of the examples are also provided, as well as the MATLAB, Python, and Julia source code to reproduce the obtained results. Finally, we present a novel phenomenon in the study of the renormalization group (RG). Namely, we found Shilnikov homoclinic orbits in the RG flow of a quantum field theory, proving the existence of chaotic RG-behavior in the vicinity of a fixed point.