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 pulse-coupled 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 our knowledge, such exotic attractors have not been reported previously, and we expect them to arise commonly in biological networks whose dynamics are governed by pulse (or spike) generation.

Original languageEnglish (US)
Pages (from-to)1-14
Number of pages14
JournalNonlinear Dynamics
DOIs
StateAccepted/In press - Apr 3 2017

Fingerprint

Attractor
Fires
Unstable
Integrate
Nonlinear dynamical systems
Dynamical systems
Symbolic Analysis
Periodic Forcing
Neuronal Network
Network Dynamics
Basin of Attraction
Biological Networks
Nonlinear Dynamical Systems
Spike
Nonlinear Dynamics
Dynamical system
Distinct

Keywords

  • Bifurcation
  • Pulse-coupled oscillators
  • Stability
  • Unstable attractor

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering

Cite this

Partially unstable attractors in networks of forced integrate-and-fire oscillators. / Zou, Hai Lin; Deng, Zi Chen; Hu, Wei Peng; Aihara, Kazuyuki; Lai, Ying-Cheng.

In: Nonlinear Dynamics, 03.04.2017, p. 1-14.

Research output: Contribution to journalArticle

Zou, Hai Lin ; Deng, Zi Chen ; Hu, Wei Peng ; Aihara, Kazuyuki ; Lai, Ying-Cheng. / Partially unstable attractors in networks of forced integrate-and-fire oscillators. In: Nonlinear Dynamics. 2017 ; pp. 1-14.
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