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Title: Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms

Abstract

We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delay time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions.

Authors:
; ;  [1]
  1. Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102 (United States)
Publication Date:
OSTI Identifier:
21464505
Resource Type:
Journal Article
Journal Name:
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics (Print)
Additional Journal Information:
Journal Volume: 82; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevE.82.036601; (c) 2010 The American Physical Society; Journal ID: ISSN 1539-3755
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BIFURCATION; COMPUTERIZED SIMULATION; CONTROL SYSTEMS; COUPLING; DIFFUSION; FAILURES; FEEDBACK; NONLINEAR PROBLEMS; ONE-DIMENSIONAL CALCULATIONS; PARTIAL DIFFERENTIAL EQUATIONS; STABILITY; TIME DELAY; TWO-DIMENSIONAL CALCULATIONS; DIFFERENTIAL EQUATIONS; EQUATIONS; SIMULATION

Citation Formats

Boubendir, Yassine, Mendez, Vicenc, Rotstein, Horacio G, Department de Fisica Grup de Fisica Estadistica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, and Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102. Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms. United States: N. p., 2010. Web. doi:10.1103/PHYSREVE.82.036601.
Boubendir, Yassine, Mendez, Vicenc, Rotstein, Horacio G, Department de Fisica Grup de Fisica Estadistica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, & Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102. Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms. United States. https://doi.org/10.1103/PHYSREVE.82.036601
Boubendir, Yassine, Mendez, Vicenc, Rotstein, Horacio G, Department de Fisica Grup de Fisica Estadistica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, and Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102. 2010. "Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms". United States. https://doi.org/10.1103/PHYSREVE.82.036601.
@article{osti_21464505,
title = {Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms},
author = {Boubendir, Yassine and Mendez, Vicenc and Rotstein, Horacio G and Department de Fisica Grup de Fisica Estadistica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona and Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102},
abstractNote = {We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delay time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions.},
doi = {10.1103/PHYSREVE.82.036601},
url = {https://www.osti.gov/biblio/21464505}, journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics (Print)},
issn = {1539-3755},
number = 3,
volume = 82,
place = {United States},
year = {Wed Sep 15 00:00:00 EDT 2010},
month = {Wed Sep 15 00:00:00 EDT 2010}
}