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August  2013, 6(4): 975-983. doi: 10.3934/dcdss.2013.6.975

Bifurcation of periodic solutions from a ring configuration of discrete nonlinear oscillators

Carlos Garca-Azpeitia 1,2,3, and  Jorge Ize 1,2,3,

1. 

Depto. Matemticas y Mecnica
2. 

IIMAS-UNAM, FENOMEC
3. 

Apdo. Postal 20-726, 01000 Mxico D.F.

Received  October 2011 Revised  April 2012 Published  December 2012

This paper gives an analysis of the periodic solutions of a ring of $n$oscillators coupled to their neighbors. We prove the bifurcation of branchesof such solutions from a relative equilibrium, and we study theirsymmetries. We give complete results for a cubic Schr?dinger potential andfor a saturable potential and for intervals of the amplitude of theequilibrium. The tools for the analysis are the orthogonal degree andrepresentation of groups. The bifurcation of relative equilibria was givenin a previous paper.
Keywords: Ring configuration, nonlinear oscillators, global bifurcation, equivariant degree theory..
Mathematics Subject Classification: 34C25, 37G40, 47H11, 54F4.
Citation: Carlos Garca-Azpeitia, Jorge Ize. Bifurcation of periodic solutions from a ring configuration of discrete nonlinear oscillators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 975-983. doi: 10.3934/dcdss.2013.6.975
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show all references

References:
[1]

AIMS Series on Differential Equations & Dynamical Systems, 1. American Institute of Mathematical Sciences (AIMS), 2006.  Google Scholar

[2]

in "Proceedings of the 3rd Conference on Localization and Energy Transfer in Nonlinear Systems" (editor, Luis Vázquez), 44-67. NJ: World Scientific, Singapore, (2003). Google Scholar

[3]

UNAM. PhD thesis, 2010. Google Scholar

[4]

Journal of Differential Equations, 251 (2011), 3202-3227. doi: 10.1016/j.jde.2011.06.021.  Google Scholar

[5]

Preprint, 2012. Google Scholar

[6]

Journal of Differential Equations, 252 (2012), 5662-5678. doi: 10.1016/j.jde.2012.01.044.  Google Scholar

[7]

in "Topological Nonlinear Analysis" Progr. Nonlinear Differential Equations Appl., 15, 341-463. Birkhäuser Boston, (1995).  Google Scholar

[8]

De Gruyter Series in Nonlinear Analysis and Applications, 8. Walter de Gruyter, Berlin, 2003. doi: 10.1515/9783110200027.  Google Scholar

[9]

J. Phys. A: Math. Gen., 37 (2004), 2201-2222. doi: 10.1088/0305-4470/37/6/017.  Google Scholar

[10]

Nonlinearity, 7 (1994), 1623-1643.  Google Scholar

[11]

Phys. Lett. A, 374 (2010), 3912-1919. doi: 10.1016/j.physleta.2010.07.022.  Google Scholar

[12]

Physica D., 238 (2009), 687-698. doi: 10.1016/j.physd.2009.01.001.  Google Scholar

[13]

Milan J. Math., 73 (2005), 103-144. doi: 10.1007/s00032-005-0040-2.  Google Scholar

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