Systems that generate solutions with small period

Abstract

The problem of existence of periodic solutions is one of the traditional problems of the theory of differential equations. In this short note, we select a class of systems of differential equations for which conditions of the existence of periodic solutions have an extremely simple form. In particular, this class includes systems that correspond to periodically perturbed equations of oscillations without friction, ¨x + f(x) = h(t); for such systems, this problem was intensively studied. Let (j1, . . . , jn) be a permutation of the set (1, . . . ,n). We consider a system of differential equations x˙ i = fi(xji ), i= 1, . . . ,n, in which any function fi is continuous on R. This system has the property of generation of solutions with a small period if for any numberM >0 there exists a number ω0 = ω0(M) > 0 such that if 0 < ω ≤ ω0 and hi(t, x1, . . . , xn) are continuous on R × Rn, ω-periodic in t functions that satisfy the inequalities |hi| ≤ M, then the system x˙ i = fi(xji) + hi(t, x1, . . . ,xn), i= 1, . . . ,n, has an ω-periodic solution. We show that a system has the property of generation of solutions with a small period if and only if the following equalities hold: fi(R) = R, i= 1, . . . ,n. It is also shown that the condition of smallness of the period is essential. Refs 5.