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Dynamics of traveling pulses in heterogeneous media

Chaos 17, 037104 (2007); doi:10.1063/1.2778553

Published 28 September 2007

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Yasumasa Nishiura
Research Institute for Electronic Science, Hokkaido University, Sapporo 060-0812, Japan

Takashi Teramoto
Chitose Institute of Science and Technology, Chitose 066-8655, Japan

Xiaohui Yuan
Research Institute for Electronic Science, Hokkaido University, Sapporo 060-0812, Japan

Kei-Ichi Ueda
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
One of the fundamental issues of pulse dynamics in dissipative systems is clarifying how the heterogeneity in the media influences the propagating manner. Heterogeneity is the most important and ubiquitous type of external perturbation. We focus on a class of one-dimensional traveling pulses, the associated parameters of which are close to drift and/or saddle-node bifurcations. The advantage in studying the dynamics in such a class is twofold: First, it gives us a perfect microcosm for the variety of outputs in a general setting when pulses encounter heterogeneities. Second, it allows us to reduce the original partial differential equation dynamics to a tractable finite-dimensional system. Such pulses are sensitive when they run into heterogeneities and show rich responses such as annihilation, pinning, splitting, rebound, as well as penetration. The reduced ordinary differential equations (ODEs) explain all these dynamics and the underlying bifurcational structure controlling the transitions among different dynamic regimes. It turns out that there are hidden ordered patterns associated with the critical points of ODEs that play a pivotal role in understanding the responses of the pulse; in fact, the depinning of pulses can be explained in terms of global bifurcations among those critical points. We focus mainly on a bump and periodic types of heterogeneity, however our approach is also applicable to general cases. It should be noted that there appears to be spatio-temporal chaos for a periodic type of heterogeneity when its period becomes comparable with the size of the pulse. ©2007 American Institute of Physics
History: Received 30 April 2007; accepted 9 August 2007; published 28 September 2007
Permalink: http://link.aip.org/link/?CHAOEH/17/037104/1
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KEYWORDS and PACS

Keywords
PACS
  • 05.45.-a
    Nonlinear dynamics and nonlinear dynamical systems
  • 02.30.Oz
    Bifurcation theory
  • 02.30.Jr
    Partial differential equations
  • 02.60.Lj
    Ordinary and partial differential equations; boundary value problems
  • YEAR: 2007

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PUBLICATION DATA

ISSN:
1054-1500 (print)   1089-7682 (online)
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