Geodesic
The Dynamical Impact of a Shortcut in Unidirectionally Coupled Rings of Oscillators
J.P. Pade1 ; L. Lücken1 ; S. Yanchuk1
1 Humboldt-University of Berlin, Institute of Mathematics Unter den Linden 6, 10099 Berlin, Germany
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 173-189.

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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.
DOI : 10.1051/mmnp/20138511

J.P. Pade 1 ; L. Lücken 1 ; S. Yanchuk 1

1 Humboldt-University of Berlin, Institute of Mathematics Unter den Linden 6, 10099 Berlin, Germany 
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J.P. Pade; L. Lücken; S. Yanchuk. The Dynamical Impact of a Shortcut in Unidirectionally Coupled Rings of Oscillators. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 173-189. doi : 10.1051/mmnp/20138511. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138511/

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