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August  2011, 30(3): 709-735. doi: 10.3934/dcds.2011.30.709

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

Norimichi Hirano 1, , Wieslaw Krawcewicz 2, and  Haibo Ruan 3,

1. 

Department of Mathematics, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama, Japan
2. 

Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080, United States
3. 

Department of Mathematics, Universität Hamburg, 20146 Hamburg, Germany

Received  February 2010 Revised  October 2010 Published  March 2011

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: periodic solutions, equivariant degree., Autonomous functional differential equations.
Mathematics Subject Classification: Primary: 34C25, 37L20, 47H11; Secondary: 34L30, 37N2.
Citation: Norimichi Hirano, Wieslaw Krawcewicz, Haibo Ruan. Existence of nonstationary periodic solutions for $\Gamma$-symmetric Lotka-Volterra type systems. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 709-735. doi: 10.3934/dcds.2011.30.709
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show all references

References:
[1]

Z. Balanov, M. Farzamirad and W. Krawcewicz, Symmetric systems of van der Pol equations, Topol. Methods Nonlinear Anal., 27 (2006), 29-90.  Google Scholar

[2]

Z. Balanov, W. Krawcewicz and H. Steinlein, "Applied Equivariant Degree," AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006.  Google Scholar

[3]

T. Bartsch, "Topological Methods for Variational Problems with Symmetries," Lecture Notes in Math., 1560, Springer-Verlag, Berlin, 1993.  Google Scholar

[4]

N. Hirano and S. Rybicki, Existence of limit cycles for coupled van der Pol equations, J. Differential Equations, 195 (2003), 194-209. doi: 10.1016/S0022-0396(03)00212-2.  Google Scholar

[5]

N. Hirano and S. Rybicki, Existence of periodic solutions for the Lotka-Volterra type systems, Nonlinear Aanalysis TMA, (2006).  Google Scholar

[6]

G. Dylawerski, K. Gęba, J. Jodel and W. Marzantowicz, An $S^1$-equivariant degree and the Fuller index, Ann. Polon. Math., 52 (1991), 243-280.  Google Scholar

[7]

K. Gęba, Degree for gradient equivariant maps and equivariant Conley index, in "Topological Nonlinear Analysis II" (Frascati, 1995), Progr. Nonlinear Differential Equations Appl., 27, Birkh\"auser Boston, Boston, (1997), 247-272.  Google Scholar

[8]

M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory Vol II," Applied Mathematical Sciences, 69, Springer-Verlag, New York-Berlin, 1988.  Google Scholar

[9]

K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Kluwer Academic Publishers, 1992.  Google Scholar

[10]

J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems," London Mathematical Society Student Texts, 7, Cambridge University Press, Cambridge, 1988.  Google Scholar

[11]

K. P. Hadeler and G. Bocharov, Where to put delays in population models, in particular in the neutral case, Canadian Applied Mathematics Quarterly, 11 (2003).  Google Scholar

[12]

G. E. Hutchinson, "An Introduction to Population Ecology," Yale University Press, New Haven, 1978.  Google Scholar

[13]

J. Ize and A. Vignoli, "Equivariant Degree Theory," de Gruyter Series in Nonlinear Analysis and Applications, 8, Walter de Gruyter & Co., Berlin-New York, 2003.  Google Scholar

[14]

R. Levins, "Evolution in Communities Near Equilibrium," in "Ecology and Evolution of Communities" (M. L. Cody and J. M. Diamond, eds.), Harvard University Press, 1975. Google Scholar

[15]

A. Biglands, Mapl$e^{TM}$ Library Package for the computations of the equivariant degree,, available at \url{http://krawcewicz.net/degree}., ().   Google Scholar

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