Differential equations with state-dependent delay: Global Hopf bifurcation and smoothness dependence on parameters.

摘要

This thesis is devoted to a few important issues in the qualitative theory of delay differential equations with a state-dependent delay. We first develop a global Hopf bifurcation theory, based on an application of the homotopy invariance of S1-equivariant degree using the formal linearization of the system at a stationary solution. Our results show that under a set of mild technical conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global Hopf continuation of periodic solutions.;We also study the second order differentiability of solutions with respect to parameters. We introduce the notion of a locally complete triple-normed linear space and obtain an extension of the well-known Uniform Contraction Principle in such a space. We then apply this principle and obtain the second order differentiability of solutions with respect to parameters in the W 1,p-norm (1 ≤ p infinity).;Keywords and phrases. Differential equation, state-dependent delay, Hopf bifurcation, homotopy invariance, locally complete space, differentiability of solution.;2000 Mathematics Subject Classification. Primary: 46B99, 46A30, Secondary: 34K05, 34K18.;We then apply our global Hopf bifurcation theory to investigate the global continuation with respect to parameters for periodic solutions. We give sufficient geometric conditions to ensure uniform boundedness of periodic solutions and obtain an upper bound of the period of periodic solutions in a connected bifurcation branch in the Fuller space. This permits us to establish the existence of fast oscillating periodic solutions.