Modulated and Localized States in a Finite Domain

A subcritical pattern-forming (Turing) instability of a uniform state, in an infinite domain, produces two branches of spatially localized states that bifurcate from the pattern-forming instability along with a uniform spatially periodic pattern. In this paper we demonstrate that branches of localized states persist as strongly amplitude-modulated patterns in large, but finite, domains with periodic boundary conditions. Our analysis is carried out for a model Swift–Hohenberg equation with a cubic-quintic nonlinearity. If the domain size exceeds a critical value, modulated states appear in secondary bifurcations from the primary branch of spatially periodic solutions. Multiple-scales analysis indicates that these secondary bifurcations occur close to the primary instability and close to the saddle-node bifurcation on the spatially periodic solution branch. As the domain size increases, extra “turns” on the snaking curve arise through a repeating sequence of saddle-node bifurcations and mode interactions be...