Polynomial Differential Equations Equivalent

As we known, it is very difficult to find the Poincarémapping for many syste-ms which are not integrated. In the 1980's the Russian mathematician Mironenko first established the theory of reflective function which is quite new method to find the Poincarémapping. In this paper we use the reflective function of Mironenko to study the fact that the perturbed differential equations are equivalent to the polyno-mial differential equations, and also consider the polynomial differential equations with continuously differentiable coefficient functions. The continuously differenti-able function F(t,x) is the reflective function of the polynomial differential equation if and only if F(t,x) satisfies the primary relation of the reflective function. If the co-efficient functions of the polynomial differential equation are 2ω-periodic funct-ions when we use the reflective function F(t,x) to study the polynomial differential equations, then the Poincarémapping of the polynomial differential equations can be expressed by T(x)=F(-ω, x). So, for any solutionφ(t;-ω,x) of the polynomial differential equations defined on [-ω,ω], it will be 2ω-periodic if and only if x is a fixed point of the mapping T, i.e, F(-ω, x)=x. If the polynomial differential equation and its perturbed differential equation have the same reflective function, then they are called equivalent. By the equivalence, If the differential equations which are periodic belong to the same equivalence class, then the Poincarémapping of the differential systems coincide, and therefore the qualitative analysis of solution of the differential systems also coincide.In this paper we study the equivalence of the polynomial differential equations and its perturbed differential equations. Particularly, when the perturbed items are the polynomial functions and the rational fractional functions, we will obtain a set of sufficient and necessary conditions of their equivalence. Therefore, our work gener-alizes and improves the research of the literature [28][29] in which the differential equations are the Riccati equations and the Able equations, respectively. Furtherm-ore, if the perturbed items are the linear combination with polynomial functions and the linear combination with rational fractional functions, we will obtain a set of suff-icient and necessary conditions of their equivalence, too. From the above facts, it can be seen that the polynomial differential equations not only equivalent to the polyno-mial differential equations, but also equivalent to the non-polynomial differential equations. That is, the Poincarémapping coincides for the periodic equations, and the qualitative analysis of solution of the differential equations also coincide. On the other hand, the analysis of the properties of solutions of the complicated differential equations can be reduced to discuss the behavior of the solutions of the singer diff-erential equations by the equivalence.In general, it is more difficult to study the perturbed differential equations than the polynomial differential equations. If we study the reflective function of the poly-nomial differential equations, then we will obtain the analysis of the properties of solutions. According to the equivalence, the analysis of the properties of solutions of the complicated differential equations can be reduced to discuss the reflective funct-ion of the singer differential equations. Thus we can know the analysis of the proper-ties of solutions of the complicated differential equations. Consequently, when we know the reflective function of a differential equation, we will get the analysis of the properties of solutions of all the differential equations which are equivalent to the differential equation. Finally, we know that a second order differential system with constant coefficients can be converted to an Able equation by the transformation and a set of linear differential equations can be converted to a Riccati equation. So it is helpful to study the analysis of the properties of solutions of the polynomial diff-erential equations for the study of the analysis of the properties of solutions of the differential systems.