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

Yassine Boubendir*

Vicenç Méndez

Horacio G. Rotstein†

  • Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102, USA

  • Department de Física Grup de Física Estadística, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

  • Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA

  • *Also at Center for Applied Mathematics and Statistics (CAMS), NJIT.
  • †Also at Center for Applied Mathematics and Statistics (CAMS), NJIT; horacio@njit.edu;.
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Phys. Rev. E 82, 036601 – Published 2 September, 2010

DOI: https://doi.org/10.1103/PhysRevE.82.036601

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.

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Phys. Rev. E 82, 036601– Published 2 September, 2010
  • Received 16 May 2010

© 2010 American Physical Society

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