# American Institute of Mathematical Sciences

October  2016, 21(8): 2601-2614. doi: 10.3934/dcdsb.2016063

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

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

Received  September 2015 Revised  April 2016 Published  September 2016

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.
Citation: Da-Peng Li. Phase transition of oscillators and travelling waves in a class of relaxation systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2601-2614. doi: 10.3934/dcdsb.2016063
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