We analyze the bifurcation diagrams of spatially localized stationary patterns that exhibit a long spatially periodic interior plateau (referred to as localized rolls). In a wide variety of contexts, these bifurcation diagrams consist of isolas or of intertwined s-shaped curves that are commonly referred to as snaking branches. These diagrams have been rigorously analyzed by connecting the existence curves of localized rolls with the bifurcation structure of fronts that connect the rolls to the trivial state. Previous work assumed that the stable and unstable manifolds of rolls were orientable. Here, we extend these results to the nonorientable case and also discuss topological barriers that prevent snaking, thus allowing only isolas to occur. The results are applied to the Swift–Hohenberg system for which we show that nonorientable roll patterns cannot snake.
Bibliographical noteFunding Information:
The authors would like to thank the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, RI, where the majority of this work was carried out as part of the 2016 Summer@ICERM program. This program also provided support for Surabhi Desai and Aric Wheeler; we acknowledge support from the NSF via the Grant DMS-1148284 that supported Melissa Stadt. Beck was partially supported by the NSF through the Grant DMS-1411460, and Sandstede was partially supported by the NSF through DMS-1408742.
- Localized patterns