Systems that generate solutions with a small period

Abstract

Let (j₁,..., j_n) be a permutation of the n-tuple (1, ..., n). A system of differential equations \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) in which each function f_i is continuous on ℝ is considered. This system is said to have the property of generation of solutions with a small period if, for any number M > 0, there exists a number ω₀ = ω₀(M) > 0 such that if 0 < ω ≤ ω₀ and h_i(t, x₁, ..., x_n) are continuous functions on ℝ × ℝⁿ ω-periodic in t that satisfy the inequalities |h_i| ≤ M the system \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) has an ω-periodic solution. It is shown that a system has the property of generation of solutions with a small period if and only if f_i(ℝ) = ℝ for i = 1,..., n. It is also shown that the smallness condition on the period is essential.