Phase transition of oscillators and travelling waves in a class of relaxation systems

Da-Peng  Li

doi:10.3934/dcdsb.2016063

Article Contents

2016, Volume 21, Issue 8: 2601-2614. Doi: 10.3934/dcdsb.2016063

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Phase transition of oscillators and travelling waves in a class of relaxation systems

Da-Peng Li^1,

1.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064

Received: August 31, 2015

Revised: March 31, 2016

Published: September 2016

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Abstract

The main purpose of this article is to investigate the phase transition of oscillation solutions and travelling wave solutions in a class of relaxation systems as follows \begin{eqnarray} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t}=\pm u(u-a)(u-b)-v+D \frac{\partial ^{2}u}{\partial^{2} x},~~~a\neq b, \\\frac{\partial v}{\partial t}=\varepsilon( mu + nv + p ), ~~~~0<\varepsilon\ll 1,\nonumber \end{array} \right. \end{eqnarray} where $a,b,m,n,p$ are parameters in this system. By using the orbit analysis method of planar dynamical system and the homoclinic bifurcation theory, the phase transitions of the solitary oscillators, kink oscillators, periodic oscillators and travelling waves in the relaxation system above are studied. Various critical parameters of the phase transition are obtained under different parametric conditions, while various sufficient conditions to guarantee the existence of the above oscillation solutions and travelling waves are given. As some applications, this paper studied the FitzHugh-Nagumo equation, the van der Pol-equation and the Winfree generic system.

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Mathematics Subject Classification: Primary: 30D05, 32H50; Secondary: 34C23, 34K18.

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