Global Dynamics Of Some Smooth And Non-Smooth Systems

Autonomous and non-autonomous differential equations have become a hot topic in research since Poincar(?)'s time. In theory, many mathematicians such as Poincar(?), Arnold and Littlewood are interested in differential equations. In engineering, many problems of many subjects such as mechanics, electronics and mathematical biology can be modeled by differential equations so that many scientists of these subjects are also interested in differential equations.First, we introduce significance of studying differential equations and three classes differential equations (a cubic Li(?)nard system, a SD oscillator and a Fil-ippov system). Meanwhile, it also introduces the research content about these differential equations and main results.Khibnik, Krauskopf and Rousseau give a global study of a cubic Li(?)nard system in (1998 Nonlinearity 18 1505-19). But they did not solve it completely since nonlocal limit cycles and homoclinic loop are unclear. Moreover, they conjecture that the double limit cycle bifurcation curve is unique. In Chapters 2 and 3, we will study this system completely for the two equilibria case and three equilibria case, respectively.Moreover, we have studied completely the two equilibria case of the cubic Li(?)nard system. First, we analyze the qualitative properties of all equilibria(including at inifnity). To study the uniqueness and nonexistence of limit cycles in some parameter regions, we give some criterions about the uniqueness and nonexistence of limit cycles of a Li(?)nard system having some equilibria. Then,in the remain parameter regions, we prove that this system has at most two limit cycles and this system has one homoclinic loop between the two homoclinic bifurcation curves. Based on the aforementioned analysis, we give the bifurcation diagram. Moreover, we give a positive answer of the two equilibria case of the conjecture of Khibnik, Krauskopf and Rousseau.Then, we have studied completely the three equilibria case of the cubic Li(?)nard system. First,we analyze the qualitative properties of all equilibria (in-cluding at inifnity). We prove that this system has at most three limit cycles and give the relative positions of bifurcation surfaces. Based on the aforementioned analysis, we give the bifurcation diagram. Moreover, we give a positive answer of the three equilibria case of the conjecture of Khibnik, Krauskopf and Rousseau.In the next two chapters, we study two classes of system with periodic excitation, where these periodic excitation are abstraction function.Further, we study the existence and uniqueness of harmonic solutions of a smooth-and-discontinuous(for short, SD) oscillator at resonance by qualitative methods. Note that the limit case of the SD oscillator is discontinuous so that some classic theory cannot applied in the limit discontinuous case.Finally, we study a class of Filippov system with small damping and s-mall periodic excitation. We get the existence and uniqueness of harmonic and n-subharmonic solutions for (?)n ? N and prove that all harmonic and n-subharmonic solutions are Lyapunov asymptotically stable. Finally, we prove that attractors of this class system are non-chaotic.