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Unstable attractors with active simultaneous firing in pulse-coupled oscillators
Hailin Zou1, Xiaofeng Gong, C-H Lai
1Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117543, Singapore.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
Summary
Unstable attractors in pulse-coupled oscillators are linked to simultaneous firing events. Perturbations splitting these events cause nearby points to leave attractors, revealing bifurcation mechanisms.
Area of Science:
- Complex Systems
- Nonlinear Dynamics
- Computational Neuroscience
Background:
- Pulse-coupled oscillators exhibit unstable attractors where nearby states diverge.
- Understanding the dynamics of these unstable attractors is crucial for complex system analysis.
Purpose of the Study:
- To identify a hidden property of unstable attractors in pulse-coupled oscillator firing event sequences.
- To elucidate the mechanism by which unstable attractors split and bifurcate.
Main Methods:
- Analysis of firing event sequences in a pulse-coupled oscillator model.
- Identification of active simultaneous firing events as a key property.
- Investigation of the effect of general perturbations on these simultaneous events.
Main Results:
- Unstable attractors in pulse-coupled oscillators are characterized by the coexistence of active simultaneous firing events.
- Perturbations that split active simultaneous firing events cause nearby points to leave the unstable attractors.
- The failure to establish active simultaneous firing events leads to the bifurcation of unstable attractors.
Conclusions:
- Active simultaneous firing events are a critical structural property underlying unstable attractors in pulse-coupled oscillators.
- The study provides a new perspective on attractor bifurcation through the lens of simultaneous firing event dynamics.
- This framework offers insights into the stability and transitions of dynamic states in oscillatory networks.
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