Topological Equivalence With Partial Hyperbolicity

Based on exponential dichotomy, generalized exponential dichotomy and fixed point theory, this thesis considers the topological equivalence of three kinds of systems. The existence of the topologically conjugated function H(t,x) of these systems are given. Some new results are obtained. This paper has four chapters.In chapter1, we briefly introduce the research background and motivation of this dissertation, and possible difficulty appears duo to prove the transformation H{t,x) is always Holder continuous (and has Holder continuous inverse). Some definitions and lemmas are also introduced in this chapter.In chapter2, We study the topological linearization of impulse nonlinear system with linear system partially satisfying IS condition.we prove that the transformation H(t,x) satisfies Holder regularity. This chapter reports an improvement of the linearization the-orem of Fenner and Pinto [21]. Fenner and Pinto showed that there exists a one-to-one correspondence between solutions of impulse linear system and impulse nonlinear sys-tem. Moreover, they aproved that H(t,x)-x is uniformly bounded, if H(t,x) denotes the transformation. However, no proof of the Holder regularity of the transformation H(t,x) appears in [21]. The main objective in this chapter is precisely to give a proof of the Holder regularity of the transformation H(t,x). Namely, we show that the topolog-ically conjugated function H(t,x) in the Hartman-Grobman theorem, is always Holder continuous (and has Holder continuous inverse). Moreover, we weakened an important assumption in [21]. Fenner and Pinto obtained the linearization theorem by setting that the whole linear system should satisfy IS condition. In this chapter, this assumption is reduced. In fact, it is enough to assume that the linear system partially satisfies IS condition.In chapter3, we consider the topological linearization of nonautonomous system with the nonlinear term is unbounded.[32] tried to reduce the nonlinear term f to unbounded case. But he made it by sacrificing to assume that the linear system should be exponentially asymptotically stable. So in this chapter, we studied the unbounded case for Palmer’s linearization theorem with dichotomy.Namely,we prove system is topologically conjugated to system.where g(t,x,y)satisfies‖g(t,x,y)‖≤δ‖y‖+M,and f(t,x,y)is arbitrary.Namely, f,g is unbounded.In chapter4,we discuss the topological equivalence of two nonlinear systems.In other words,we prove system x(t)=A(t)z(t)+f(t,x,y)is topologically conjugated to system y(t)=A(t)y(t)+t(t,x,y),where g(t,x,y)and f(t,x,y)is bounded.