Equivalence Of Differential System With Reflecting Function Is Separated Of Variable

As is well known, the properties of solutions of differential system x′= X (t , x) provide important basis for the law of motion of objects. However, the results of the research in this area are actually few and it is even difficult for the research of the properties of solutions of periodic differential system. Even though Lyapunov and Poincaréprovided some methods to study this question, still could not break through the limitation that depends on the expression of the solution. Then, in 1980s, a Russian mathematician Mironenko first established the reflecting function theory which provided a new way for the research.The reflecting function gives a new method to research the Poincarémapping of periodic differential system. After more than 20 years'research, the experts of differential equation had already obtained a lot of new good conclusions and these results provide a new theoretical basis and criterion for explaining the complex law of motion of some objects.The Russian mathematician Mironenko studied the properties of the differential system with its reflecting function is separated of variables. Based on this, using the theory of the reflecting function we mainly research the equivalence class of the given differential system. By establishing the equivalence of this kind of differential system, the study of complex differential system is reduced to researching the simple equivalent differential system of the given system.Firstly, using the formula that comes from the lemma and the fundamental formula of reflecting function, we derive the conditions with the same monotone matrix between two differential systems. Secondly, suppose the differential system is simple system, by assuming its reflecting function, the definition of simple system and the known conditions are used to proof the reflecting function of the given system is also the reflecting function of differential system fundamental formula of the reflecting function. So we obtain the equivalence of the two differential systems.The last, we study the equivalence of the given differential system and its perturbed differential system. Using the two lemmas and the Cauchy existence and uniqueness theorem we get the result. Meanwhile, we promote this result and obtain a more general conclusion.Furthermore, some examples are cited to prove the upper conclusions true.