The Topological Classification Of The Quartic Invariant Algebraic Curve Cycles In A Class Of Qudratic System

The present paper is devoted to classifying topologically to the quartic invariant algebraic curves in qudratic system,some corresponding conditions that the connected component of the quartic curve will be a homoclinic cycle of the system have been brought forward.This paper is separated into three chapters.In the first chapter ,the background and preparative knowledge of this paper is introduced.In the second chapters,we discuss the invariant algebraic curve of qudratic system. Using the topological invariability of the structure and phase portrait determined by system (1) under the action of Lie group G,we simplify the qudratic system which have homoclinic cycle which goes through the hyperbolic saddle. At the same time,the quartic invariant algebraic curves are devided into five cases with different topological structure in infinite place by the tangent cases of the curve and the infinite singular point.Then,we find all cases of the quartic invariant algebraic curves that their connected component will be a homoclinic cycle,including a two-parameter curve and fifteen one-parameter curves.Finally,we obtain the figures of all kind of curves in Poincáre disk by analyzing the quality of them.In the third chapter,we analyze the qudratic system with quartic invariant algebraic curve by qualitative method,and the corresponding global phase is drawn. Furthermore,we obtain same sufficient and necessary condition such that the homoclinic cycle of the system is defined by the quartic invariant algebraic curve. At last,appling with the rotated vector theory,we make a disturbance with parameter δ to the homoclinic bifurcation system again to arise limit cycle, as a result ,with the parameter changing the cycle decreases and finally disappears to a fine focus.The calculation of this chapter is completed by Mathematic software.