Canonical Form And Global Phase Portrait Of Planar Quintic And Sextic Quasi-homogeneous Polynomial Systems

This paper mainly investigates the canonical forms and global topological structures of planar quintic and sextic quasi-homogeneous but non-homogeneous coprime polynomial differential system, and applies inverse integrating factor to deduce the explicit expression of first integral of quintic systems. It has four chapters in this paper.In the first chapter we mainly introduce some problems of planar quasi-homogeneous system which has been studied in recent years, such as, integrability, center and focus,canonical form, and limit cycle. The second chapter introduces the basic concepts of planar quasi-homogeneous system, quasi-homogeneous blow up technique and Poincare-Lyapunov compactification, and some important lemmas used in this paper.The third chapter is devoted to the study of planar quintic quasi-homogeneous but non-homogeneous polynomial systems. Doing appropriate linear transformations, we firstly give the canonical forms of these systems which can have 0,1,2,4 parameters. Then, we analyze the qualitative structure of a neighbourhood of the unique finite singularity to obtain the local phase portrait of these canonical systems by means of quasi-homogeneous blow up method. Moreover, we use Poincare-Lyapunov compactification to analyze the property of the infinite singularities. Therefore, combining the property of finite and infinite singularities and invariant curve, we obtain all the global phase portraits of these canonical systems. We finally perform a topological classification for the set of these global phase portraits and get 52 topological equivalence classes. In addition, we also compute the first integral expression of these quintic quasi-homogeneous canonical systemsIn the forth chapter we consider the planar sextic quasi-homogeneous polynomial system.We firstly apply the algorithm given by Belen Garcia et al. [2] to get the expression of these systems, and then obtain their canonical forms by doing appropriate linear transformations.Finally, according to the canonical forms of these systems, we directly draw a conclusion that it has neither center, focus nor limit cycle in planar sextic quasi-homogeneous but non-homogeneous porime polynomial differential system.