Yanchuk S. Bifurcation theory for singularly perturbed systems with delay and applications to synchronization problems

In this thesis, foundations of the bifurcation theory for singularly perturbed differential equations with delay are developed. For systems of differential-difference equations, new synchronization effects are discovered and investigated. The bifurcation theory includes the following elements: stability analysis of an equilibrium and spectral properties of linearized systems, classification of local bifurcations, derivation of amplitude equations, which play the role of normal forms in the vicinity of a bifurcation, estimation of the error between solutions of the amplitude equations and the corresponding exact solutions of the delay system. The obtained results allow connecting singularly perturbed delay differential equations with other classes of systems, such as partial differential equations or differential-difference equations. The universality of the Eckhaus destabilization mechanism for high-dimensional systems is shown. Further, high-dimensional systems of ordinary and delay differential equations are studied. The synchronization problem for symmetrically coupled ordinary differential equations is shown to be equivalent to the stability problem for some linear invariant subspace. Necessary and sufficient conditions for synchronization are obtained using Lyapunov functions method. In the case of nonidentical systems, the obtained estimates show robustness with respect to the symmetry breaking perturbations. Eigenvalue spectrum of equilibria and Floquet exponents of periodic solutions are studied for high-dimensional differential-difference equations with the rotational coupling symmetry. The obtained results are applied in order to investigate dynamics of a system of two optically coupled semiconductor lasers in detail.