Partially unstable attractors in networks of forced integrate-and-fire oscillators

Abstract

The asymptotic attractors of a nonlinear dynamical system play a key role in the long-term physically observable behaviors of the system. The study of attractors and the search for distinct types of attractor have been a central task in nonlinear dynamics. In smooth dynamical systems, an attractor is often enclosed completely in its basin of attraction with a finite distance from the basin boundary. Recent works have uncovered that, in neuronal networks, unstable attractors with a remote basin can arise, where almost every point on the attractor is locally transversely repelling. Herewith we report our discovery of a class of attractors: partially unstable attractors, in pulsecoupled integrate-and-fire networks subject to a periodic forcing. The defining feature of such an attractor is that it can simultaneously possess locally stable and unstable sets, both of positive measure. Exploiting the structure of the key dynamical events in the network, we develop a symbolic analysis that can fully explain the emergence of the partially unstable attractors. To

Wireless-Coupled Oscillator Systems With an Injection-Locking Signal

“…As stated, the analysis of [14]- [15] is limited to coupled systems in which none of the oscillator components is subject to an injection-locking source, as widely considered in previous research on distributed synchronization of sensor networks. However, the works [25]- [27] address coupled systems with a reference timing signal, introduced into one or more oscillator elements, which can be easily done through injection locking or phase locking. A reference signal with a high spectral purity will reduce the phase noise in all the system oscillators, due to the wireless coupling.…”

Attractors with controllable basin sizes from cooperation of contracting and expanding dynamics in pulse-coupled oscillators

“…[12] Another interesting type of attractor, i.e., partially unstable attractors, with both stable and unstable points can also appear in pulsecoupled oscillators under periodic forcing. [13] Unstable attractors are observed in pulse-coupled oscilla-tors with delays, where the delays account for the transmission time of the pulses. [14][15][16][17][18][19] Such models can be used to study various natural phenomena, such as neural dynamics where pulses correspond to the generations of spikes, and synchronization of fireflies where pulses are the flashes.…”