Applications of Topological and Perturbation Methods to Analysis of Periodic Solutions in Delay Differential Equations: Classification of Symmetries, Asymptotic Approximation and Stabilization
The present dissertation contains several interconnected results regarding periodic solutions to delay differential equations (DDEs). First of all, we formulate and prove a theorem that guarantees the occurrence of the Hopf bifurcation of relative periodic solutions from a relative equilibrium in a general Γ × S 1 - equivariant system of functional differential equations (FDEs) using the method based on twisted equivariant degree with one free parameter. This theorem also allows to classify the symmetries of the relative periodic solutions. The theoretical result is illustrated through the series of examples including D8- and S5-symmetric coupling of identical mode-locked semiconductor lasers and D8-symmetric configuration of coupled electro-mechanical oscillators with hysteresis. The latter example shows the possibility to adapt the proposed method for the settings with weakened conditions on the smoothness. Secondly, we perform the analysis of a rather broad class of slow-fast delayed models of population dynamics, that exhibit the behavior similar to the aforementioned mode-locked semiconductor lasers. In particular, we study the mechanism of formation of pulsating periodic solutions as well as develop a nonlocal method for their asymptotic approximations. Finally, we develop a noninvasive delay feedback (Pyragas) control to make a neutrally stable periodic orbit of a Hamiltonian system exponentially stable. More specifically, we establish different sufficient conditions for the stabilization of orbits with small and large amplitudes. We also present a discussion of how these conditions agree with each other.