The Geometric Analysis Of One Kind Of Singularly Perturbed Problem

Differential equations or systems with a small parameter before the highest derivative have been the research hotspot of dynamic system,and gradually developed into an important branch.When the small parameter tends to 0 or even takes 0,the decrease of order of equations leads to the essential changes of the structure and corresponding dynamic behavior of the equation,which leads to a singularly perturbated problem,and this kind of problems contain differente scales often.In the early stage of development,the focus is the asymptotic analysis of bound-ary value problems.1970 s,the groundbreaking work of N.Fenichel established the basis of ge-ometric singular perturbation.Then the research focus of singular perturbation theory shifts to geometric singular perturbation gradually.However,Fenichel's theory has good results only for the case of normally hyperbolic,and fails for non-normally hyperbolic point or man-ifold.In two-dimensional space,a common case of non-normally hyperbolic occurs when the attractive critical manifold intersects with the repulsive critical manifold,such intersection is called a turning point.Such non-normally hyperbolic point appears in the classical Van der Pol equation and Lienard equation.Another common case of non-normally hyperbolic occurs when the critical manifold intersects itself at the turning point,and there are several stable manifolds and unstable manifolds near the turning point usually.This phenomenon is generally called exchange of stabilityIn our paper,we use the blow-up method to study the phenomenon of exchange of sta-bility.Fenichel's theory guarantees the existence of the invariant manifolds near the normally hyperbolic manifold.For the non-normally hyperbolic point,locally,we study the turning point of the system after standardization.In the process of standardizing the equation,we found there appears another parameter before the small parameter naturally,then it leads to a singularly perturbed branch problem.We study the dynamic behavior of the standardized branch system under different parameters by using blow-up method and obtain some results.Then we use some examples to verify the correctness of our conclusion.