TY - JOUR

T1 - Systems that generate solutions with a small period

AU - Pilyugin, S. Yu.

AU - Rodioniva, A. A.

N1 - Pilyugin, S.Y., Rodionova, A.A. Systems that generate solutions with a small period. Vestnik St.Petersb. Univ.Math. 49, 256–259 (2016). https://doi.org/10.3103/S1063454116030109

PY - 2016/9/30

Y1 - 2016/9/30

N2 - 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.

AB - 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.

KW - system of differential equations

KW - periodic solution

U2 - https://doi.org/10.3103/S1063454116030109

DO - https://doi.org/10.3103/S1063454116030109

M3 - Article

VL - 49

SP - 256

EP - 259

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -