American Institute of Mathematical Sciences

Advanced
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

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

show all references

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

[1]

Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985
[2]

Zalman Balanov, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree, part I: An axiomatic approach to primary degree. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 983-1016. doi: 10.3934/dcds.2006.15.983
[3]

Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263
[4]

Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387
[5]

Ruediger Landes. Stable and unstable initial configuration in the theory wave fronts. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 797-808. doi: 10.3934/dcdss.2012.5.797
[6]

Wenxian Shen. Global attractor and rotation number of a class of nonlinear noisy oscillators. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 597-611. doi: 10.3934/dcds.2007.18.597
[7]

R. Yamapi, R.S. MacKay. Stability of synchronization in a shift-invariant ring of mutually coupled oscillators. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 973-996. doi: 10.3934/dcdsb.2008.10.973
[8]

Tetsutaro Shibata. Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102
[9]

Zalman Balanov, Meymanat Farzamirad, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 923-960. doi: 10.3934/dcds.2006.16.923
[10]

Xiaoxue Zhao, Zhuchun Li, Xiaoping Xue. Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring. Networks & Heterogeneous Media, 2018, 13 (2) : 323-337. doi: 10.3934/nhm.2018014
[11]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[12]

Dan-Andrei Geba, Manoussos G. Grillakis. Large data global regularity for the classical equivariant Skyrme model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5537-5576. doi: 10.3934/dcds.2018244
[13]

Chun-Hsiung Hsia, Chang-Yeol Jung, Bongsuk Kwon. On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3319-3334. doi: 10.3934/dcdsb.2018322
[14]

Johanna D. García-Saldaña, Armengol Gasull, Hector Giacomini. Bifurcation values for a family of planar vector fields of degree five. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 669-701. doi: 10.3934/dcds.2015.35.669
[15]

Juan Sánchez, Marta Net, José M. Vega. Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1357-1380. doi: 10.3934/dcdsb.2006.6.1357
[16]

Gunog Seo, Gail S. K. Wolkowicz. Pest control by generalist parasitoids: A bifurcation theory approach. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020163
[17]

Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503
[18]

Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. A minimal approach to the theory of global attractors. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2079-2088. doi: 10.3934/dcds.2012.32.2079
[19]

Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701
[20]

Klaus Reiner Schenk-Hoppé. Random attractors--general properties, existence and applications to stochastic bifurcation theory. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 99-130. doi: 10.3934/dcds.1998.4.99

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences