Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T10:15:28.538Z Has data issue: false hasContentIssue false
  • English
  • Français

The Dynamical Impact of a Shortcut in Unidirectionally Coupled Rings of Oscillators

Published online by Cambridge University Press:  17 September 2013

J.P. Pade ,
L. Lücken  and
S. Yanchuk
Get access

Abstract

We study the destabilization mechanism in a unidirectional ring of identical oscillators, perturbed by the introduction of a long-range connection. It is known that for a homogeneous, unidirectional ring of identical Stuart-Landau oscillators the trivial equilibrium undergoes a sequence of Hopf bifurcations eventually leading to the coexistence of multiple stable periodic states resembling the Eckhaus scenario. We show that this destabilization scenario persists under small non-local perturbations. In this case, the Eckhaus line is modulated according to certain resonance conditions. In the case when the shortcut is strong, we show that the coexisting periodic solutions split up into two groups. The first group consists of orbits which are unstable for all parameter values, while the other one shows the classical Eckhaus behavior.

Keywords

network perturbationcoupled oscillatorssymmetry breakingshortcutunidirectional ring
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 173 - 189
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abrams, D. M., Strogatz, S. H.. Chimera states for coupled oscillators. Phys. Rev. Lett., 93 (2004), 174102. CrossRefGoogle ScholarPubMed
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwanga, D.-U.. Complex networks: Structure and dynamics. Phys. Rep., 424 (2006), 175–308. CrossRefGoogle Scholar
Bressloff, P. C., Coombes, S., de Souza, B.. Dynamics of a ring of pulse-coupled oscillators: Group-theoretic approach. Phys. Rev. Lett., 79 (1997), 2791–2794. CrossRefGoogle Scholar
Collins, J.J., Stewart, I.. A group-theoretic approach to rings of coupled biological oscillators. Biol. Cybern., 71 (1994), 95–103. CrossRefGoogle ScholarPubMed
Daido, H.. Strange waves in coupled-oscillator arrays: Mapping approach. Phys. Rev. Lett., 78 (1997), 1683–1686. CrossRefGoogle Scholar
E. J. Doedel. AUTO-07P: Continuation and bifurcation software for ordinary differential equations. Montreal, Canada, April 2006.
W. Eckhaus. Studies in Non-Linear Stability Theory, vol. 6 of Springer Tracts in Natural Philosophy. Springer, New York, 1965.
Fenichel, N.. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21 (1971), 193–226. CrossRefGoogle Scholar
Horikawa, Y., Kitajima, H.. Transient chaotic rotating waves in a ring of unidirectionally coupled symmetric bonhoeffer-van der pol oscillators near a codimension-two bifurcation point. Chaos, 22 (2012), 033115. CrossRefGoogle Scholar
Horikawa, Y.. Exponential dispersion relation and its effects on unstable propagating pulses in unidirectionally coupled symmetric bistable elements. Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 2791–2803. CrossRefGoogle Scholar
Koseska, A., Kurths, J.. Topological structures enhance the presence of dynamical regimes in synthetic networks. Chaos, 20(4):045111, 2010. CrossRefGoogle ScholarPubMed
Y. Kuznetsov. Elements of Applied Bifurcation Theory. vol. 112 of Applied Mathematical Sciences. Springer-Verlag, 1995.
Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., Alon, U.. Network motifs: Simple building blocks of complex networks. Science, 298 (2002), 824–827. CrossRefGoogle ScholarPubMed
M. Pecora, L., Carroll, T. L.. Master stability functions for synchronized coupled systems. Phys. Rev. Lett., 80 (1998), 2109–2112. CrossRefGoogle Scholar
Perlikowski, P., Yanchuk, S., Popovych, O. V., Tass, P. A.. Periodic patterns in a ring of delay-coupled oscillators. Phys. Rev. E, 82 (2010), 036208. CrossRefGoogle Scholar
Perlikowski, P., Yanchuk, S., Wolfrum, M., Stefanski, A., Mosiolek, P., Kapitaniak, T.. Routes to complex dynamics in a ring of unidirectionally coupled systems. Chaos, 20 (2010), 013111. CrossRefGoogle Scholar
Popovych, O. V., Yanchuk, S., Tass, P. A.. Delay- and coupling-induced firing patterns in oscillatory neural loops. Phys. Rev. Lett. 107 (2011), 228102. CrossRefGoogle ScholarPubMed
Restrepo, J. G., Ott, E., Hunt, B. R.. Desynchronization waves and localized instabilities in oscillator arrays. Phys. Rev. Lett., 93 (2004), 114101. CrossRefGoogle ScholarPubMed
Restrepo, J. G., Ott, E., Hunt, B. R.. Spatial patterns of desynchronization bursts in networks. Phys. Rev. E, 69 (2004), 066215. CrossRefGoogle ScholarPubMed
Strelkowa, N., Barahona, M.. Transient dynamics around unstable periodic orbits in the generalized repressilator model. Chaos, 21 (2011), 2011. CrossRefGoogle ScholarPubMed
Takamatsu, A., Tanaka, R., Yamada, H., Nakagaki, T., Fujii, T., Endo, I.. Spatiotemporal symmetry in rings of coupled biological oscillators of physarum plasmodial slime mold. Phys. Rev. Lett., 87 (2001), 078102. CrossRefGoogle Scholar
Tuckerman, L. S., Barkley, D.. Bifurcation analysis of the Eckhaus instability. Physica D, 46 (1990), 57–86. CrossRefGoogle Scholar
Van der Sande, G., Soriano, M. C., Fischer, I., Mirasso, C. R.. Dynamics, correlation scaling, and synchronization behavior in rings of delay-coupled oscillators. Phys. Rev. E, 77 (2008), 055202. CrossRefGoogle Scholar
Vishwanathan, A., Bi, G., Zeringue, H.C.. Ring-shaped neuronal networks: a platform to study persistent activity. Lab Chip, 11 (2011), 1081–8. CrossRefGoogle Scholar
Waller, I., Kapral, R.. Spatial and temporal structure in systems of coupled nonlinear oscillators. Phys. Rev. A, 30 (1984), 2047–2055. CrossRefGoogle Scholar
Yanchuk, S., Wolfrum, M.. Destabilization patterns in chains of coupled oscillators. Phys. Rev. E, 77 (2008), 026212. CrossRefGoogle ScholarPubMed
Zou, W., Zhan, M.. Splay states in a ring of coupled oscillators: From local to global coupling. SIAM J. Appl. Dyn. Syst., 8 (2009), 1324–1340. CrossRefGoogle Scholar