Math. Model. Nat. Phenom.
Volume 8, Number 5, 2013Bifurcations
|Page(s)||173 - 189|
|Published online||17 September 2013|
The Dynamical Impact of a Shortcut in Unidirectionally Coupled Rings of Oscillators
Humboldt-University of Berlin, Institute of Mathematics Unter den
Linden 6, 10099
⋆ Corresponding author. E-mail: firstname.lastname@example.org
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.
Mathematics Subject Classification: 34C15 / 34C25 / 34C23 / 34D06 / 34D10 / 37C75
Key words: network perturbation / coupled oscillators / symmetry breaking / shortcut / unidirectional ring
© EDP Sciences, 2013
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