Limit cycles in the presence of convection, a travelling wave analysis

TL;DR

This paper analyzes limit cycles in a convection-influenced diffusion model using a travelling wave approach, transforming the equations into a polar form and conducting numerical experiments to understand wave propagation and asymmetry.

Contribution

It introduces a transformation method for irregular reaction-diffusion equations with convection and applies travelling wave analysis to the Schnakenberg model.

Findings

Traveling wave speed differs from initial estimates.

System exhibits asymmetric behavior on either side of the wave.

Numerical experiments support the analytical insights.

Abstract

We consider a diffusion model with limit cycle reaction functions, in the presence of convection. We select a set of functions derived from a realistic reaction model: the Schnakenberg equations. This resultant form is unsymmetrical. We find a transformation which maps the irregular equations into model form. Next we transform the dependent variables into polar form. From here, a travelling wave analysis is performed on the radial variable. Results are complex, but we make some simple estimates. We carry out numerical experiments to test our analysis. An initial `knock' starts the propagation of pattern. The speed of the travelling wave is not quite as expected. We investigate further. The system demonstrates distinctly different behaviour to the left and the right. We explain how this phenomenon occurs by examining the underlying behaviour.