Abstract

Joint presentation by Edgardo Villa-Sepulveda and Alan Champneys, Department of Engineering Mathematics, University of Bristol.   Within spatially extended pattern formation systems, so-called snakes and ladders bifurcation diagrams of localised patterns arise within so called Pomeau pinning regions. These patterns can be understood qualitatively using the geometric theory of dynamical systems. In the limit of a certain codimension-two bifurcation Turing bifurcation, these complex bifurcation diagrams are known to arise within an exponentially thin parameter wedge around a Maxwell equal-energy curve.  In principle, the coefficient of this beyond-all-orders envelope can be calculated in specific examples, using a theory pioneered by Kozyreff and Chapman for the case of the Swift-Hohenberg equation. The purpose of this talk is to explore how far this theory can be generalised to general pattern formation models, such as the Schakenberg and Brusselator reaction-diffusion systems. We shall also discuss related beyond-all-orders problems that arise in dynamical systems, such as the local birth of chaos from a saddle-node-Hopf bifurcation in low-order dissipative dynamical systems.