Study On Periodic Solutions Of A Class Of Periodic Differential Equations

In this paper,we investigate the period and number of periodic solutions of a class of piecewise polynomial periodic differential equations.where x,?,??R.k? E Z and 0<?<?,f_n and g_m are polynomials of degree n and m,respectively.The differential equation is closely related to the periodic abel equation,and is widely used in the mathematical modeling of some ecological problems.The main results of this paper are as follows:1)We generalize the basic theory of differential equations to the piecewise C~r periodic differential equations,such as the existence and uniqueness of solution,continuity,differentiability and analyticity of solutions on the initial values and parameters,and so on.2)The period of periodic solutions of this equation is ? if this equation has periodic solutions.3)The maximum number H(n,m)of isolated periodic solutions of piecewise polynomial differential equations is at least max{m,n}.Especially,when 0<m,n ?3 and(m,n)?(3,3)or m=n=3 but coefficients of the highest order term of polynomial f_n(x)and g_m(x)have the same sign,H(n,m)is max{m,n}.4)The number of isolated periodic solutions of piecewise polynomial dif-ferential equations are finite,if some conditions of number n,m and polynomial f_n(x),g_m(x)are satisfied.5)We discuss the bifurcation of periodic solutions of a class of differential equations which describes the harvesting of species.where h is a harvesting rate.