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January  1999, 5(1): 61-82. doi: 10.3934/dcds.1999.5.61

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, Germany
2. 

Department of Mathematics, Pennsylvania State University, P.O. Box PSU, Lehman, PA 18627, United States

Received  September 1997 Revised  October 1998 Published  October 1998

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.

Keywords: Symmetric periodic solutions, floquet multipliers, bifurcations..
Mathematics Subject Classification: 34.
Citation: Peter Dormayer, Anatoli F. Ivanov. Stability of symmetric periodic solutions with small amplitude of $\dot x(t)=\alpha f(x(t), x(t-1))$. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 61-82. doi: 10.3934/dcds.1999.5.61
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