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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, 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 |
References:
[1] |
Z. Balanov, M. Farzamirad and W. Krawcewicz, Symmetric systems of van der Pol equations, Topol. Methods Nonlinear Anal., 27 (2006), 29-90. |
[2] |
Z. Balanov, W. Krawcewicz and H. Steinlein, "Applied Equivariant Degree," AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006. |
[3] |
T. Bartsch, "Topological Methods for Variational Problems with Symmetries," Lecture Notes in Math., 1560, Springer-Verlag, Berlin, 1993. |
[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. |
[5] |
N. Hirano and S. Rybicki, Existence of periodic solutions for the Lotka-Volterra type systems, Nonlinear Aanalysis TMA, (2006). |
[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. |
[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. |
[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. |
[9] |
K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Kluwer Academic Publishers, 1992. |
[10] |
J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems," London Mathematical Society Student Texts, 7, Cambridge University Press, Cambridge, 1988. |
[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). |
[12] |
G. E. Hutchinson, "An Introduction to Population Ecology," Yale University Press, New Haven, 1978. |
[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. |
[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. |
[15] |
A. Biglands, Mapl$e^{TM}$ Library Package for the computations of the equivariant degree, available at http://krawcewicz.net/degree. |
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. |
[2] |
Z. Balanov, W. Krawcewicz and H. Steinlein, "Applied Equivariant Degree," AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006. |
[3] |
T. Bartsch, "Topological Methods for Variational Problems with Symmetries," Lecture Notes in Math., 1560, Springer-Verlag, Berlin, 1993. |
[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. |
[5] |
N. Hirano and S. Rybicki, Existence of periodic solutions for the Lotka-Volterra type systems, Nonlinear Aanalysis TMA, (2006). |
[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. |
[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. |
[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. |
[9] |
K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Kluwer Academic Publishers, 1992. |
[10] |
J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems," London Mathematical Society Student Texts, 7, Cambridge University Press, Cambridge, 1988. |
[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). |
[12] |
G. E. Hutchinson, "An Introduction to Population Ecology," Yale University Press, New Haven, 1978. |
[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. |
[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. |
[15] |
A. Biglands, Mapl$e^{TM}$ Library Package for the computations of the equivariant degree, available at http://krawcewicz.net/degree. |
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