Homoclinic snaking & localised patterns beyond all asymptotic orders #

Edgardo Villar-Sepúlveda, Alan Champneys

14:50 Monday in 2Q49.

Part of the Dynamics of reaction-transport systems session.

Abstract #

Lots of research has come up after Alan Turing’s breakthrough relating a stabilizing process with the destabilization of a spatially homogeneous steady state. Since then, many things have been said about the so-called Turing bifurcations that occur naturally in reaction-diffusion equations. These facts have taken different directions. Some authors have provided conditions to have these kinds of bifurcations., Others have analysed the criticality of the bifurcations, whilst others have studied conditions under which one can ensure the appearance of homoclinic snaking around a homogeneous-steady state near the bifurcation point. In this talk, I am going to explain a general mechanism to study the latter in any reaction-diffusion equation fulfilling general conditions. My work is based on a study carried out by Chapman and Kozyreff looking at this from another perspective. In particular, they analyzed this for the Swift-Hohenberg equation, which can be seen as a degenerate reaction-diffusion equation, so their research is useful as a starting point for the study of conditions for the appearance of snaking in general reaction-diffusion equations.