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Periodic patterns in a ring of delay-coupled oscillators.
P Perlikowski1, S Yanchuk, O V Popovych
1Institute of Mathematics, Humboldt University of Berlin, 10099 Berlin, Germany.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
Summary
Coupling delays in oscillator rings create new rotating waves by splitting existing ones. Increasing oscillator numbers can replace these delay effects, impacting neural network models.
Area of Science:
- Nonlinear dynamics
- Complex systems theory
- Computational neuroscience
Background:
- Spatiotemporal periodic patterns, such as rotating waves, are crucial in various natural phenomena.
- Understanding the stability and formation of these patterns in coupled systems is a key challenge.
- Delayed couplings introduce complex dynamics not present in instantaneous coupling models.
Purpose of the Study:
- To investigate the impact of coupling delays on rotating waves in unidirectional oscillator rings.
- To analyze the mechanisms underlying the splitting of rotating waves due to delays.
- To explore alternative methods for controlling or replicating the effects of coupling delays.
Main Methods:
- Analysis of Hopf bifurcations in systems with delayed couplings.
- Numerical simulations using Stuart-Landau oscillators.
- Modeling of coupled FitzHugh-Nagumo systems representing neural networks with chemical synapses.
Main Results:
- Delayed couplings induce the splitting of single rotating waves into multiple new ones.
- The onset of rotating waves is governed by Hopf bifurcations from a symmetric equilibrium.
- Increasing the number of oscillators can effectively substitute for the influence of coupling delays.
Conclusions:
- Coupling delays significantly alter the behavior of rotating waves in oscillator networks.
- The number of oscillators is a critical parameter that can modulate the system's dynamics, mimicking delay effects.
- Findings are applicable to understanding neuronal network synchronization and pattern formation.
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