ON THE PERIODIC SOLUTIONS OF THE RATIONAL DIFFERENTIAL EQUATIONS

To see this let us note that the phase curves of (0.2) near the origin (0,0) in polar coordinates cos , sin x r y r θ θ = = are determined by (0.3), where ( ), ( ), 0,1, 2, , i i a b i n θ θ = ... are polynomials in cos , θ sin . θ The limit cycles of (0.2) correspond to 2π -periodic solutions of (0.3). The planar vector field (0.2) has a center at (0,0) if and only if the equation (0.3) has a center at 0, r = i. e., all the solutions nearby are 2π -periodic [1]–[3]. The method of Lyapunov is often used to study the center-focus problem, but for high-order systems it is very difficult to give the center conditions. In this paper the method of reflecting function to study the behavior of solutions of (0.1) with the sufficient conditions for r = 0 to be a center is applied. Now the concept of the reflecting function, which will be used throughout the rest of this article is introduced.