Let (j 1,..., j n ) be a permutation of the n-tuple (1, ..., n). A system of differential equations x˙=fi(xji),i=1,…,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, x 1, ..., x n ) are continuous functions on ℝ × ℝn ω-periodic in t that satisfy the inequalities |h i | ≤ M the system x˙=fi(xji),i=1,…,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.
Original languageEnglish
Pages (from-to)256-259
Number of pages4
JournalVestnik St. Petersburg University: Mathematics
Volume49
Issue number3
DOIs
StatePublished - 30 Sep 2016

    Scopus subject areas

  • Mathematics(all)

    Research areas

  • system of differential equations, periodic solution

ID: 52509970