Pitchfork bifurcation along a slow parameter ramp: Coherent structures in the critical scaling

Author:

Goh Ryan1,Kaper Tasso J.1,Scheel Arnd2

Affiliation:

1. Department of Mathematics and Statistics Boston University Boston Massachusetts USA

2. School of Mathematics University of Minnesota Minneapolis Minnesota USA

Abstract

AbstractWe investigate the slow passage through a pitchfork bifurcation in a spatially extended system, when the onset of instability is slowly varying in space. We focus here on the critical parameter scaling, when the instability locus propagates with speed , where is a small parameter that measures the gradient of the parameter ramp. Our results establish how the instability is mediated by a front traveling with the speed of the parameter ramp, and demonstrate scalings for a delay or advance of the instability relative to the bifurcation locus depending on the sign of , that is on the direction of propagation of the parameter ramp through the pitchfork bifurcation. The results also include a generalization of the classical Hastings–McLeod solution of the Painlevé‐II equation to Painlevé‐II equations with a drift term.

Funder

National Science Foundation

Publisher

Wiley

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