The Analysis Of Relaxation Oscillations In Several Singular Perturbation Systems By Matching Asymptotic Expansion

By using matching asymptotic expansion and geometric singular perturbation theory, this thesis is mainly concerned with the existence of relaxation oscillations (including the classical and the canard ones) in several singular perturbation system-s. Also we forcus on the applications of the theoretical results to several predator-prey systems with large parameter(s) and get the existence of relaxation oscillations as well as the corresponding parameter conditions in these models. The thesis is divided into four chapter.Chapter 1 is the introduction, in which, the background and the motivation of the researches in this thesis are stated. Also, we review the singular perturbation methods and introduce the main works and the methods utilized in this paper.In chapter 2. by using matching asymptotic expansion, we focus on the birth of relaxation oscillations in generalized singular perturbation Lienard equation. It is shown that relaxation oscillation and maximal canard oscillation can be detected by using matching asymptotic expansion, in which, only the zeroth-order equations of the outer and inner expansions are needed. It is also shown that the order of turning-points plays an essential role in the birth of relaxation oscillations. Relaxation oscillations can occur only when the order of turning points is odd, and otherwise, relaxation oscillations are impossible if there is turning point of even order. Finally, a van der Pol-Duffing oscillations with large damping is taken as an illustrating example to verify the main results.In chapter 3, we study the fast-slow dynamics in a biological predator-prey system with a large parameter. Firstly, we define the singular saddles and the singular nodes by combing the fast and the slow orbits of the degenerated and the layer systems. Then based on these, we analyze the types of the equilibrium points. According to the different positions of the positive equilibrium point, we show the existence of relaxation oscillations and canard relaxation oscillations on the basis of matches asymptotic expansion and geometric singular perturbation theory. Finally, the theoretical results are verified by numerical simulations.In chapter 4, we study the existence of relaxation oscillations and canard re-laxation oscillations in a three-dimensional singular perturbation system. By the projecting the dynamics of this three-dimensional system into a suitable family of planes, we then apply the method in Chapter 2 to get the existence of classical and canard relaxation oscillations in the planes. By doing so, we get the cylinders consisting of the classical and canard relaxation oscillations. Finally, the existence of classical and canard relaxation oscillations in the three-dimensional singular per-turbation svstem follows from the Poincare-Bendixson Theorem.