Existence of nonstationary periodic solutions for $\Gamma$-symmetricLotka-Volterra type systems

Norimichi Hirano; Wieslaw Krawcewicz; Haibo Ruan

doi:10.3934/dcds.2011.30.709

Article Contents

2011, Volume 30, Issue 3: 709-735. Doi: 10.3934/dcds.2011.30.709

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Existence of nonstationary periodic solutions for $\Gamma$-symmetric Lotka-Volterra type systems

1.
Department of Mathematics, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama
2.
Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080
3.
Department of Mathematics, Universität Hamburg, 20146 Hamburg

Received: January 31, 2010

Revised: September 30, 2010

Published: August 2011

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Abstract

In this paper we present a general framework for applications of the twisted equivariant degree (with one free parameter) to an autonomous $\Gamma$-symmetric system of functional differential equations in order to detect and classify (according to their symmetric properties) its periodic solutions. As an example we establish the existence of multiple non-constant periodic solutions of delay Lotka-Volterra equations with $\Gamma$-symmetries. We also include some computational examples for several finite groups $\Gamma$.

Keywords:

Autonomous functional differential equations,
periodic solutions,
equivariant degree.

Mathematics Subject Classification: Primary: 34C25, 37L20, 47H11; Secondary: 34L30, 37N25.

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