Homoclinic snaking in bounded domains

Abstract

Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states. However, with non-Neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of "snaking without bistability", recently observed in simulations of binary fluid convection by Mercader, Batiste, Alonso and Knobloch (preprint).

Spatial localization in heterogeneous systems

“…16(a) and 16(b)]. Once again the resulting behavior resembles closely that familiar from systems exhibiting localized states with non-Neumann boundary conditions [25,30]. Fig.…”

“…In each case we applied multiplicative spatial forcing by allowing the parameter a to depend on space. Forcing of this type preserves the homogeneous state u = 0 while selecting preferred locations for the spatial structures, somewhat in the manner of finite domain boundary conditions [25,26]. We first examined the effects of periodic forcing with wavelength equal to the natural wavelength generated by the Swift-Hohenberg equation.…”

“…As it does so, a defect is created that flattens the central region creating a state reminiscent of a two-pulse state [5,24]. The resulting behavior resembles that of localized structures in SH23 on nonperiodic domains with mixed boundary conditions [25,26].…”

“…[25] for a system where like symmetries are destroyed through the imposition of mixed (or Robin) boundary conditions. Depending on the sign of α, two types of localized solutions may be produced.…”

Localized rotating convection with no-slip boundary conditions

“…40,41 More generally, multipulse states are found in the snaking region of 1-pulse states, both in model equations like the Swift-Hohenberg equation 42 and in convection problems without a conserved quantity. 28 In these systems, however, the 1-pulse and 2-pulse states remain distinct, and no connections between them are present.…”