Complex spatiotemporal behavior in  a chain of one-way nonlinearlycoupled elements

Yuri B. Gaididei; Rainer Berkemer; Carlos Gorria; Peter L. Christiansen; Atsushi Kawamoto; Takahiro Shiga; Mads P. Sørensen; Jens Starke

doi:10.3934/dcdss.2011.4.1167

Article Contents

2011, Volume 4, Issue 5: 1167-1179. Doi: 10.3934/dcdss.2011.4.1167

This issue Previous Article Nonlinear lattice models for biopolymers: Dynamical coupling to a ionic cloud and application to actin filaments Next Article Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations

Complex spatiotemporal behavior in a chain of one-way nonlinearly coupled elements

1.
Bogolyubov Institute for Theoretical Physics, Metrologichna str. 14 B, 01413, Kiev
2.
Department of Mathematics, Technical University of Denmark, DK-2800 Kongens Lyngby
3.
Department of Applied Mathematics and Statistics, University of the Basque Country, E-48080 Bilbao
4.
Department of Informatics and Mathematical Modeling & Department of Physics, Technical University of Denmark, DK-2800 Kongens Lyngby
5.
Toyota Central R&D Labs, Inc., Nagakute, 480-1192 Aichi
6.
Department of Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby

Received: August 31, 2009

Revised: December 31, 2009

Published: September 2011

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Abstract

The dynamics of asymmetrically coupled nonlinear elements is considered. It is shown that there are two distinctive regimes of oscillatory behavior of one-way nonlinearly coupled elements depending on the relaxation time and the strength of the coupling. In the subcritical regime when the relaxation time is shorter than a critical one a spatially uniform stationary state is stable. In the supercritical regime due to a Hopf bifurcation traveling waves spontaneously create and propagate along the system. Our analytical approach is in good agreement with numerical simulations of the fully nonlinear model.

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Mathematics Subject Classification: Primary: 34K18, 35C07, 35B36; Secondary: 37C75.

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