On The Qualitative Study Of A Planar Singularly Perturbed System Whose Critical Manifold Has Degenerate Extreme Point

This paper aims to study the qualitative theory of a planar singularly perturbed system whose critical manifold has degenerate extreme point as the following: Where F(x)=x/4+x/7,0<ε＜＜1,αis a control parameter.This paper is divided into three chapters and organized as follows:In chapter 1, we briefly present the development, main results of singular Lienard systems and geometric theory and bifurcation theory of singularly perturbed systems. Especially, we present the study situation about duck solutions and duck cycles nowadays.In chapter 2, we mainly study the equilibrium point of system (1) when a is a parameter which is independent withε. Especially, we give the wholly qualitative analysis of Lienard system (1) withα= 0 and prove that the solution of (1) is bounded on the whole surface. Moreover, we get the existence of relaxation oscillation by constructing the inner and outer boundary curve. Then, we get the whole Poincare phase diagram.In chapter 3, we mainly consider the existence of duck solution of system (1) withα=α(ε). By WKB method, we get the implicit solution. Then we get the equation aboutαaccording to definition of duck solution and method of asymptotic analysis. Therefore, we get the asymptotic expansion of parameterα。Finally, we explain the following phenomenon by using Matlab mathematical software:1) Whenεis big, there doesn't exist so-called duck phenomenon (we get the upper bound ofεis 1/64). When s turns small and a doesn't change, the explosive position becomes closer and closer to the extreme point if the system still holds the duck phenomenon;2) Whenα∈(-1,0) is close to the extreme point andεdoesn't change, the explosive point rises up according to the y-axis if the system still holds the duck phenomenon, which means the trajectory goes a longer distant along the unstable manifold.