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

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April 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

Abstract
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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

References:

[1]	Z. Balanov, W. Krawcewicz and H. Steinlein, "Applied Equivariant Degree,", AIMS Series on Differential Equations & Dynamical Systems, (2006). Google Scholar
[2]	J. C. Eilbeck and M. Johansson, The discrete nonlinear Schrödinger equation - 20 years on,, in, (2003), 44. Google Scholar
[3]	C. García-Azpeitia, "Aplicación del Grado Ortogonal a la Bifurcación en Sistemas Hamiltonianos,", UNAM. PhD thesis, (2010). Google Scholar
[4]	C. García-Azpeitia and J. Ize, Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators,, Journal of Differential Equations, 251 (2011), 3202. doi: 10.1016/j.jde.2011.06.021. Google Scholar
[5]	C. García-Azpeitia and J. Ize, Global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the n-body problem,, Preprint, (2012). Google Scholar
[6]	C. García-Azpeitia and J. Ize, Bifurcation of periodic solutions from a ring configuration in the vortex and filament problems,, Journal of Differential Equations, 252 (2012), 5662. doi: 10.1016/j.jde.2012.01.044. Google Scholar
[7]	J. Ize, Topological bifurcation,, in, 15 (1995), 341. Google Scholar
[8]	J. Ize and A. Vignoli, "Equivariant Degree Theory,", De Gruyter Series in Nonlinear Analysis and Applications, 8 (2003). doi: 10.1515/9783110200027. Google Scholar
[9]	M. Johansson, Hamiltonian Hopf bifurcations in the discrete nonlinear Schrödinger trimer: oscillatory instabilities, quasiperiodic solutions and a 'new' type of self-trapping transition,, J. Phys. A: Math. Gen., 37 (2004), 2201. doi: 10.1088/0305-4470/37/6/017. Google Scholar
[10]	R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or hamiltonian networks of weakly coupled oscillators,, Nonlinearity, 7 (1994), 1623. Google Scholar
[11]	P. Panayotaros, Continuation of normal modes in finite NLS lattices,, Phys. Lett. A, 374 (2010), 3912. doi: 10.1016/j.physleta.2010.07.022. Google Scholar
[12]	C. L. Pando and E. J Doedel, Bifurcation structures and dominant models near relative equilibria in the one-dimensional discrete nonlinear Schrödinger equation,, Physica D., 238 (2009), 687. doi: 10.1016/j.physd.2009.01.001. Google Scholar
[13]	S. Rybicki, Degree for equivariant gradient maps,, Milan J. Math., 73 (2005), 103. doi: 10.1007/s00032-005-0040-2. Google Scholar

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References:

[1]	Z. Balanov, W. Krawcewicz and H. Steinlein, "Applied Equivariant Degree,", AIMS Series on Differential Equations & Dynamical Systems, (2006). Google Scholar
[2]	J. C. Eilbeck and M. Johansson, The discrete nonlinear Schrödinger equation - 20 years on,, in, (2003), 44. Google Scholar
[3]	C. García-Azpeitia, "Aplicación del Grado Ortogonal a la Bifurcación en Sistemas Hamiltonianos,", UNAM. PhD thesis, (2010). Google Scholar
[4]	C. García-Azpeitia and J. Ize, Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators,, Journal of Differential Equations, 251 (2011), 3202. doi: 10.1016/j.jde.2011.06.021. Google Scholar
[5]	C. García-Azpeitia and J. Ize, Global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the n-body problem,, Preprint, (2012). Google Scholar
[6]	C. García-Azpeitia and J. Ize, Bifurcation of periodic solutions from a ring configuration in the vortex and filament problems,, Journal of Differential Equations, 252 (2012), 5662. doi: 10.1016/j.jde.2012.01.044. Google Scholar
[7]	J. Ize, Topological bifurcation,, in, 15 (1995), 341. Google Scholar
[8]	J. Ize and A. Vignoli, "Equivariant Degree Theory,", De Gruyter Series in Nonlinear Analysis and Applications, 8 (2003). doi: 10.1515/9783110200027. Google Scholar
[9]	M. Johansson, Hamiltonian Hopf bifurcations in the discrete nonlinear Schrödinger trimer: oscillatory instabilities, quasiperiodic solutions and a 'new' type of self-trapping transition,, J. Phys. A: Math. Gen., 37 (2004), 2201. doi: 10.1088/0305-4470/37/6/017. Google Scholar
[10]	R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or hamiltonian networks of weakly coupled oscillators,, Nonlinearity, 7 (1994), 1623. Google Scholar
[11]	P. Panayotaros, Continuation of normal modes in finite NLS lattices,, Phys. Lett. A, 374 (2010), 3912. doi: 10.1016/j.physleta.2010.07.022. Google Scholar
[12]	C. L. Pando and E. J Doedel, Bifurcation structures and dominant models near relative equilibria in the one-dimensional discrete nonlinear Schrödinger equation,, Physica D., 238 (2009), 687. doi: 10.1016/j.physd.2009.01.001. Google Scholar
[13]	S. Rybicki, Degree for equivariant gradient maps,, Milan J. Math., 73 (2005), 103. doi: 10.1007/s00032-005-0040-2. Google Scholar

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