Reflective Function Of The Cubic Polynomial Differential Systems

As is well known, in the objective world the motion of many objects can be described by differential system The study of the regularity of this motion needs only to discuss the property of the solutions of system (*).If X (t + 2ω, x ) = X (t , x) (ωis a positive constant), to study the solution's behavior of system (*), we could use, as introduced in [7], Poincarémapping. But it is very difficult to seek Poincarémapping for many systems which are not integrated in finite terms. In 1980's the Russian mathematician Mironenko [9] first established the theory of reflective function (RF). It permits one to find and use symmetries of differential systems with respect to several variables (in particular, time symmetries related with the replacement of t by -t ). Since then a quite new method to study system (*) has been established.Assume that X (t , x ) is a continuously differential function on R×Rn, and that exists a unique solution for the initial problem of system (*). If x = -( t ; t₀ , x₀) is a solution of system (*) satisfying x (t₀ )= x0, then its reflective function can be defined by F ( t , x ) = ?( - t ; t , x). Hence if X (t + 2ω, x ) = X (t , x), then the Poincarémapping of system (*) can be expressed by [9]: T ( x ) = F ( -ω, x). So if an arbitrary solution x (t ) of system (*) exists on interval [ -ω,ω], it will be 2kω- periodic solution, if and only if, when x := x ( - kω)is a solution of equation F ( - kω, x )= x. It is a new task to apply the reflective function to study the property of the solutions of the periodic system, and there are many problems to be studied.In this paper , we study the expression of F₂ (t , x , y ) when F (t , x , y ) = ( x , F₂ (t , x , y))T is the reflective function of the cubic variable coefficient polynomial differential system (where ai (t ), b_j(t ) are continuously differentiable functions for t∈R). And we obtain a good result of F₂ (t , x , y )= f₂₀ (t , x ) + f₂₁(t , x )y. By applying this result, we establish the necessary and sufficient conditions of the reflect function F (t , x , y ) = ( x , F₂ (t , x , y))T of the system. We also get the 2ω?periodic system's Poincarémapping and properties of the periodic solution. As a feature of this paper, we generalize the conclusions of the quadratic polynomial differential system in the papers [18][22]. Finally, some examples are cited to prove the upper conclusions true. There conclusions play an important role in the applications of biological mathematics and control theory.