And Autonomy Of Quadratic Polynomial Differential System Equivalent Non-autonomous Polynomial Differential System

As is well known,to discuss the propertities of solutions of the differential equation x′=X(t ,x) is very important for deep theory value and practical value when study the law of motion of these objects in the objective world. In normal circumstances, it is difficult to study the geometry of its solution ,and the geometric solution for the autonomous system x′=X(t ,x) of state research has made a number of fruitful results by the efforts of experts and scholars at home and abroad, particularly in the plane polynomial systems exist limit cycles,stability has a lot of good results. The famous Professor Ye Yanqian has published two monographs in this direction. For non-autonomous differential system is relatively limited research findings, and which has been for the periodic system of the famous Lyapunov transformation and Poincarémapping.Mironenko also created a new way that get its Poincarémapping by the method of reflective function,thus discussing the propertities of solutions will be achieved .With the same reflective function of differential system which will be called equivalent,equivalent system class of periodic solutions have the same state. Therefore,to study a class of non-linear differential equation of state,just research that the system is equivalent to a linear system or the behavior of solutions of autonomous system can be.This article is based on the existing literature [ 27? 43] on how to construct non-autonomous differential system equivalent to autonomous differential system for further study.In the introduction,we introduce the article research background, present status, research significance. In the prior knowledge, to convenient it gives a detailed definition and basic properties of the reflection function,reflective function and Poincarémapping theorem.These concepts throughout the text from beginning to end.Known from the literature [1 ]of Ye Yanqian,any autonomous second order polynomial differential systems can be turned into a totalⅢclass system:And a number of well-known results have been achieved on the solutions of this typeⅢclass system of geometric properties.As a main part of the article,firstly,we studied the centered second order polynomial differential system(1),and construct the following non-autonomous polynomial differential system equivalent to it based on the necessary and sufficient conditions of the equivalence.Moreover, general non-autonomous differential system equivalent to autonomous differential system was constructed according to the general integral of the system (1). The Hamilton system was also discussed with the same method and ideas.Finally, we give one example to verify the correctness of the conclusion above.