Qualitative Analysis Of A Kind Of Plane Quartic Polynomial Differential Systems

In this paper, we discuss a certain plane quartic system with stellar node: In study, we obtain the global phase portraits of the system, its corresponding algebraic conditions and some results about its structural stability.So far, systematic results about this system is still out of reach, which is due to so many parameters and higher order critical points. In order to overcome these difficulties, not only do we use qualitative theory and analytical method, we also use bifurcation method and the idea of algebraic classification of system.This system does not have limit cycles. The approach handling the system is characteristic polynomial. In this paper, we discuss fifth-order binary forms. In the first chapter, through the discussion of the fifth-order binary forms and the use of qualitative theory and bifurcation methods, we obtain the theorem of infinite critical point and the theorem of finite critical point. By the way, there is something deserves to be mentioned: by means of transcritical bifurcation, we get a special type of critical point. Here is a short description about transcritical bifurcation: The transcritical bifurcation only happens when the system has an equilibrium that exists for all values of the parameter and can never be destroyed. When this equilibrium collides with another equilibrium, the two equilibria exchange their stability property, but continue to exist both before and after the bifurcation. Hence, the two equilibria "pass through each other".In chapter II, first, we give a concise account of the theory of algebraic invariant. Second, by using the methods of algebraic invariants, we get the classifications of fifth-order binary forms. At last, we get the normal forms of the system(0.1) with different kinds of characteristic polynomials.In order to achieve a better understanding of global structure of the system, first, we introduce a new concept that is called double sector region in chapter III. Then according to the lemmas in chapter I, we get two lemmas called lemma 3.1, lemma 3.2, and one theorem on the structure of the system. The system (0.1) has and only has 6 topological different double sector regions, which constitute the phase portait of the system. On the basis of the theorem about the structure of the system, we obtain 121 topologically different global phase portraits.In chapter IV, we illustrate number systems of system I-XII, which give a further proof to the theorem on the structure of the system and 121 kinds of global phase portraits. By the way, there is something deserves to be mentioned: sometimes the process of illustration is rather difficult, so we apply to MATLAB in order to improve efficiency and increase precision. Due to the limitation of the length of this article, we only offer 24 examples.On the basis of results in previous chapters, we get the sufficient and necessary conditions for the structural stability of the system.Theorem 5.1 (sufficient and necessary conditions for structural stability of the system) For system I, it is structurally stable if and only if P1,P2,P3,P4,P5≠0.For system III, it is structurally stable if and only iff;For system IV, it is structurally stable if and only if P1,P2,P3 ≠0 .For system V, it is structurally stable if and only if P1≠0.Theorem 5.2 System II, VI, VII, VIII, IX, X, XI and XII are structurally unstable.