Research On The Focus And Even Equivalence Of Differential System

The research of qualitative behavior of solutions of differential system, it not only plays an important role in the development of the theory of differential equations, at the same time in the study of moving objects in the objective world, biological population changes, the universe, satellite trajectory, complex network dynamics also has a great practical value. The differential system can be divided into two categories:autonomous system and non-autonomous system. The research progress of the qualitative behavior of solutions of the autonomous system is closely related to the Hilbert’s sixteenth problem, namely the study of the center focus problem.At present, although a variety of methods have been tried, only the center focus problems of two order polynomial systems and some special cubic polynomial systems have been solved. However, the center focus problems for general cubic polynomial systems or higher order polynomial systems are still unsolved. This paper will adopt a new method, Mironenko reflective function method to research center focus problems of some cubic systems. The third chapter will primarily transform the cubic polynomial systems into rational fractional differential equations, which are periodic. Then we will study the necessary and sufficient conditions for the rational fractional differential equations that have linear reflective functions and rational fractional reflective functions. Then we can get some sufficient conditions for the cubic system that whose origin is a center by using the given conclusions. In addition, studying the qualitative behavior for a time-varying system is very difficult generally. If we can give a qualitative relationship to build up a complex differential system with a simple differential system, then the behavior of solutions of the simple system can determine the behavior of solutions of the complex system. For example, Mironenko has established the equivalence, the odd-even equivalence and ω-equivalence between two differential systems and so on. Using these equivalence properties, the qualitative behavior of solutions of differential systems can be studied more conveniently.In this paper, due to previous research, the even equivalence between the time-varying Abel equation and autonomous equations, will be studied detailedly. Many necessary and sufficient conditions have been established for the Abel equation, which is even equivalent to a given stationary equation; the obtained results are applied to discuss the qualitative behaviors of the periodic solutions of these equations.