Researches On Several Non-smooth Dynamical Systems

In this thesis, we study several non-smooth dynamical systems, including the impulsive difference equations and the contrast structure solutions in some non-smooth singularly perturbed equations. The whole thesis is divided into the following chapters.Chapter one reviews the history and the development of the singular pertur-bation and non-smooth dynamical system, introduces some basic concepts and theorems relevant to our study, and gives the main research work and innovative points for this thesis.Chapter two and three study a class of impulsive difference equations from a multiscale point of view. Since impulsive differential equations are subjected to abrupt changes in their states, the solutions to these equations are piecewise continuous in general, then the study for these equations becomes difficult. By applying the ideas of singular perturbation and introducing the suitable singu-larly perturbed terms, these Chapters extend the original impulsive differential equations into the singularly perturbed problem with infinite initial values and the singularly perturbed boundary value problem in the critical case respectively. Then the continuous/smooth asymptotic solutions to these correlative problems are constructed to describe the discontinuous solutions of the impulsive differen-tial equations effectively. Moreover, the uniform valid of the asymptotic solutions is proved.Chapter four is devoted to investigate the contrast structure solution for the second order semi-linear non-smooth singularly perturbed equation. By trans-forming the right-hand function of the equation at some point t, the types of equilibrium points to the additional equation are changed. By doing so, there can exist the spike-type and step-type contrast structure solutions in the left and right interval respectively. Then the possible types of the solution are given in the whole interval with the phase-plane analysis. By using the method of boundary function, we construct the asymptotic expressions of the spike-type and step-type solutions, also determine the impulsive and transition point. At point t, we make the solution smoothly by sewing. Finally the asymptotic solution is shown and proved to be uniformly valid in the whole interval.Chapter five discusses the contrast structure solution for the quasi-linear non-smooth singularly perturbed equation with slow variable. The asymptotic solution of this problem is constructed by the boundary function method. By sewing the orbit smoothly, the existence of the contrast structure solution is shown and the asymptotic solution is proved to be uniformly valid in the whole interval.Chapter six considers the contrast structure solutions for non-smooth sin-gularly perturbed equations with slow-fast layers, including the slow-fast layers happen in the different points and the slow-fast layers happen in the same point. By introducing the different scales in the slow-fast layers, the asymptotic solutions are constructed in two intervals. By sewing the orbit smoothly, the asymptot-ic solutions for these problems are presented and demonstrated to be uniformly valid in the whole interval. The remainder estimation also is derived.Finally, the thesis summaries our work, and points out the future research directions.