Quasi-periodic Solutions For Differential Equations With State-dependent Delay

In this dissertation,we study the quasi-periodic solutions and their analyticity for differential equations with state-dependent delays(SD-DDE).In the light of parameterization method,we transform the evolution problem concerning the ex-istence of quasi-periodic solutions to some functional equations in Banach spaces,to which various methods in nonlinear analysis can be applied.Furthermore,the results are given in an a-posteriori format,i.e.given an approximate solution sat-isfying some non-degenerate condition,there would be a true solution nearby.The dissertation is organized as follows.In Chapter 1,we first,introduce the fundamental theory of the solutions for differential equations with state-dependent delay.Then we give a brief introduction to some established results on the analyticity of solutions for delay differential equation.At last,we summarize some recent results on the quasi-periodic solutions for delay differential equations.This chapter servers as a general background for the dissertation.In Chapter 2,under the hyperbolic assumption,we prove the persistence of quasi-periodic solutions for SD-DDE in the finitely differentiable framework.To overcome the lack of regularity of the composition operator,we develop a fixed point theorem by the interpolation inequality and prove the existence of quasi-periodic solution in an a-posteriori format.It is worthy noticing that the small divisor problem does not appear under the hyperbolic assumption.The method in this chapter also works in the case of multiple state-dependent delays.In Chapter 3,we give a brief introduction to foliation-preserving torus maps,which enables us to investigate the analyticity problem of quasi-periodic solutions for SD-DDE.By analyzing the resonance of frequency vector,we show the simple dynamics of rotation torus maps which preserve the foliation.Furthermore,we employ the KAM techniques to the conjugation problem of foliation-preserving torus maps in the non-perturbative setting.In Chapter 4,we study the existence of analytic quasi-periodic solutions of a class of SD-DDE.To control the varying analytic domain caused by the state-dependent delay,we introduce another functional equation to solve the invariance equation induced by the parameterization method.It turns out that the auxiliary equation comes from the conjugation problem of delay mapping,which preserves the foliation.We apply the KAM techniques to solve the two coupled functional equations and prove that,on some parameter set with positive Lebesgue measure,there exist analytic quasi-periodic solutions satisfying local uniqueness.Finally,we conclude the results above and propose a conjecture on the an-alyticity and non-analyticity of quasi-periodic solution for SD-DDE.In addition,we show a few potential projects in the future work.