To Snake or Not to Snake in the Planar Swift–Hohenberg Equation

Abstract

We investigate the bifurcation structure of stationary localised patterns of the two-dimensional SwiftHohenberg equation on an infinitely long cylinder and on the plane. On cylinders, we find localised roll, square and stripe patches that exhibit snaking and non-snaking behaviour on the same bifurcation branch. Some of these patterns snake between four saddle-node limits: recent analytical results predict then the existence of a rich bifurcation structure to asymmetric solutions, and we trace out these branches and the PDE spectra along these branches. On the plane, we study the bifurcation structure of fully localised roll structures, which are often referred to as worms. In all the above cases, we use geometric ideas and spatial-dynamics techniques to explain the phenomena we encounter.

Localized rotating convection with no-slip boundary conditions

“…Recent work on these systems has identified the presence of spatially localized convection in the form of structures that have been called convectons [5][6][7][8][9][10] and the presence of such structures has been related to the phenomenon of homoclinic snaking that has been extensively studied in the context of model systems such as the Swift-Hohenberg equation. [11][12][13][14] These new states consist of a finite number of rolls embedded in a background conduction state and are located in the so-called snaking or pinning region in parameter space. Within this region the convectons lie on a pair of intertwined branches: in systems with midplane symmetry these consist of states of odd and even parity; when this symmetry is absent the branches consist of even parity states with either downflow or upflow in their center.…”

Nonsnaking doubly diffusive convectons and the twist instability

“…States of this type expand abruptly in the extended direction near a special point in parameter space corresponding to the formation of a pair of heteroclinic connections between two different fixed points in a spatial dynamics view of the system. 26 This point, variously referred to as the nonsnaking 35 or protosnaking 36 point plays the role of a Maxwell point in systems, like the Swift-Hohenberg equation (1), with gradient dynamics. If the spatial eigenvalues of one of the fixed points are complex the resulting behavior may be termed collapsed snaking.…”

Snaking bifurcations in a self-excited oscillator chain with cyclic symmetry

“…The snaking structure has e.g. been studied a lot in the one dimensional [16] and the two dimensional [17] SwiftHohenberg equation, which is a convenient and generic model system to study fundamental properties of the arising dynamics.…”

“…Notice that the higher the energy of the solution, the larger the number of convection cells. Often the two snaking branches are also interconnected through a number of unstable branches, and a ladder like pattern emerges [17].…”

“…In many respects this is quite surprising, since non-linear oscillators with subcritical Hopf bifurcations, often coupled to neighbouring oscillators of the same type into chains or arrays, are actually very common models for a number of systems from engineering vibrations. And also the appearance of bi-or multi-stability, which is obviously at the core of the phenomenon [13][16] [17], is well established in many of these engineering systems. Moreover, the emergence of spatially localised vibration states in structural dynamics is also a well known observational fact: e.g.…”