Isolas Versus Snaking of Localized Rolls

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.