Research On Triangular Differential Systems With The Same Reflection Function

Since Mironenko[1]created the reflection function of a differential system,many experts have used this theory to study qualitative behaviors of the solution of a differential system.In particular,when the reflection function of a differential system is known and the system is 2?-periodic,the Poincare mapping can be established by means of the reflection function,so that the number of periodic solutions and stability of the periodic system are solved.By using the equivalence of differential systems,we can obtain that the behavior of periodic solutions of a periodic differential system which is equivalent to the periodic system.Unfortunately,it is very difficult to find a reflection function for a differential system in general.Hence it is a very interesting question to determine the equivalence of two systems in the case of unknown reflection function.For the differential system in[7],Mironenko has given if A(t,x)satisfies then the perturbation system is equivalent to system(1),where the a(t)is the odd scalar function of t.Hence,the result has implied that the system is also equivalent to system(1),where a1(t)is the odd scalar function and A(t,x)is the solution of(2).It can be seen that the solution of(2),which is the reflection integral is particularly important to determine the equivalence of two differential systems.In[31],Bel'skii has presented the structural forms of the reflection integrals for Riccati equation and the Abel equation and the general polynomial equation and have obtained sufficient conditions such that there exist reflection integrals for these equations.Bel'skii have studied in[32},the equivalence of the quadratic differential system and the quadratic triangular differential system in view of the reflection integral with the quadratic polynomial form,and investigated solution behavior of the general time-varying quadratic polynomial differential system by using the behavior of this system.In this paper,basing on previous researches,we apply the Mironenko's reflection function method to explore the structural form of the reflection integral for the cubic triangular differential system and the equivalence relation with the corresponding perturbation differential system.Firstly,we focus on the equivalence conditions for the general cubic differential system and as well as the characteristics of Aij(t)and Bij(t).To be more specific,in view of the reflection integral with the cubic polynomial type of the differential system(4),we discuss the structural form of reflection integral for system(4).Then the sufficient conditions are obtained to ensure there exists the reflection integral with these Structured form for system(4).Fuethermore,we discuss the necessary conditions for the equivalence of(3)and(4),and the behavior of their periodic solutions when(3)and(4)are periodic systems in t.Secondly,we also study the equivalent conditions for(3)and the differential system:and then obtain the characteristics of(3),in which case(3)need not be a triangular system,on this occasion,we get the structural form of the reflection integral and the sufficient conditions for reflection integrals with these structural form of(4).