Unstable attractors induce perpetual synchronization and desynchronization

Abstract

Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which \textit{unstable attractors} arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.

14 pages, 12 figures

References in corpus (2)

• Slow Switching in Globally Coupled Oscillators: Robustness and Occurrence through Delayed Coupling
• Prevalence of unstable attractors in networks of pulse-coupled oscillators

Cited by in corpus (19)

• Mathematical frameworks for oscillatory network dynamics in neuroscience
• Topological Speed Limits to Network Synchronization
• Stable Irregular Dynamics in Complex Neural Networks
• Controlling Recurrent Neural Networks by Conceptors
• Long Chaotic Transients in Complex Networks
• Unstable Attractors: Existence and Robustness in Networks of Oscillators With Delayed Pulse Coupling
• Designing the Dynamics of Spiking Neural Networks
• The simplest problem in the collective dynamics of neural networks: Is synchrony stable?
• Speed of synchronization in complex networks of neural oscillators Analytic results based on Random Matrix Theory
• Designing Complex Networks
• Speed of complex network synchronization
• Controlled Perturbation-Induced Switching in Pulse-Coupled Oscillator Networks
• Bifurcation From Networks of Unstable Attractors to Heteroclinic Switching
• Lyapunov-like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems
• Reconfigurable Computation in Spiking Neural Networks
• Dynamical formation of stable irregular transients in discontinuous map systems
• Isochronous Dynamics in Pulse Coupled Oscillator Networks with Delay
• Partially unstable attractors in networks of forced integrate-and-fire oscillators
• Stochastic facilitation in heteroclinic communication channels