Discussion On The Consistency Of The Reflection Function Between The Three-dimensional Non-autonomous Differential System And The Autonomous Linear System

When people study the law of change of things in the objective world,they often come down to the qualitative behaviors study of the solution of a non-autonomous non-linear differential model.However,due to the complexity of the model and the feasibility of the research methods,few results of the behaviors of the solutions of non-autonomous nonlinear differential model have been achieved.We know that for autonomous,especially for autonomous polynomial differential systems,after years of unremitting efforts by experts and scholars at home and abroad,a lot of good results have been achieved.However,it is even more difficult to study the behavior of solutions for non-autonomous polynomial differential systems,especially for high-dimensional non-autonomous polynomial differential systems,so the good results are not much.Can we use the autonomous system to study the behavior of non-autonomous systems?We know that Poincare transfonmation can transform the study of periodic time-varying systems into the study of autonomous systems.In general,the establishment of transformations is very difficult.Mironenko’s reflective approach provides a new way to establish Poincare transformations.It can be used to study the behavior of solutions of non-autonomous systems,especially the qualitative behavior of periodic solutions can be directly transformed into the qualitative behavior of solutions of autonomous systems.In this paper,through Mironenko’s reflection function theory,the necessary and sufficient conditions for the three-dimensional non-autonomous nonlinear differential systems(?)are studied respectively.In addition,the structure of X,Y,Z is also studied in this paper,especially when X,Y,Z are quadratic polynomials,the necessary and sufficient conditions and expressions of their equivalence are discussed.YWhat9s more,the conclusions of the application are used to study the qualitative behavior of the periodic solutions of these three-dimensional non-autonomous systems.