Stability of symmetric periodic solutions with small amplitude of $\dot x(t)=\alpha f(x(t), x(t-1))$

Peter Dormayer; Anatoli F. Ivanov

doi:10.3934/dcds.1999.5.61

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1999, Volume 5, Issue 1: 61-82. Doi: 10.3934/dcds.1999.5.61

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Stability of symmetric periodic solutions with small amplitude of $\dot x(t)=\alpha f(x(t), x(t-1))$

Peter Dormayer^1, and
Anatoli F. Ivanov^2,

1.
Mathematisches Institut der Universität Giessen, Arndtstrasse 2, 35392 Giessen
2.
Department of Mathematics, Pennsylvania State University, P.O. Box PSU, Lehman, PA 18627

Received: August 31, 1997

Revised: September 30, 1998

Published: December 1998

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Abstract

We study special symmetric periodic solutions of the equation

$\dot x(t) =\alphaf(x(t), x(t-1))$

where $\alpha$ is a positive parameter and the nonlinearity $f$ satisfies the symmetry conditions $f(-u, v) = -f(u,-v) = f(u, v).$ We establish the existence and stability properties for such periodic solutions with small amplitude.

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Mathematics Subject Classification: 34K.

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Stability of symmetric periodic solutions with small amplitude of $\dot x(t)=\alpha f(x(t), x(t-1))$

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