The Swift-Hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states. The presence of these states is related to a phenomenon called homoclinic snaking. Numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states. The evolution of the localized states once they lose stability is illustrated using direct simulations in time. I. INTRODUCTION Ever since the observation that the subcritical complex Ginzburg-Landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states. The presence of these states has important consequences for other systems described by partial differential equations on the line since the Ginzburg-Landau equation describes the behavior of such systems near bifurcation from the trivial state of the system. Specifically, the complex Ginzburg-Landau equation describes the evolution of a long wavelength oscillatory instability, as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry. In contrast, near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real Ginzburg-Landau equation, and this equation possesses only unstable spatially localized states. It is of interest therefore to examine what happens to these unstable states at larger amplitude, where the real Ginzburg-Landau equation no longer provides an adequate description of the system. In this paper we show that the localized states can become stable at such amplitudes, and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions. We are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1:1 resonance, and use this theory to construct a large number of such states. The stability properties of these states are also determined, and the evolution of nonstationary localized states is studied by numerical integration in time. It is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation. The theory was originally developed in the context of water waves, where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ODE. The resulting localized states are called solitary waves, and in some cases turn out to be solitons. Kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area. Specifically, the ODE is viewed as a dynamical system in space, and localized states are sought as homoclinic orbits connecting the trivial state to itself. Whether such orbits are possible depends in part on the stability properties of the trivial state: eigenvalues with positive real part indicate that a nontrivial state can grow from x=, while eigenvalues with negative real part indicate that such a state may return, under appropriate conditions, back to the trivial state as x →. The spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system. In many cases, and in particular in the case considered here, the ODE is reversible. As a result the bifurcations that are encountered as a parameter is increased are nongeneric. In the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1:1 resonance. The unfolding of this resonance has been worked out in detail by Iooss and Peroueme 3, and can be used to understand the appearance of a variety of homoclinic orbits in this system, and hence of localized states with different spatial structure.

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Localized states in the generalized Swift-Hohenberg equation.

We study a partial integro-differential equation defined on a spatially extended domain that arises from the modeling of "working" or short-term memory in a neuronal network The equation is capable of supporting spatially localized regions of high activity which can be switched "on" and "off" by transient external stimuli We analyze the effects of coupling between units in the network, showing that if the connection strengths decay monotonically with distance, then no more thanon e regionof high activity canpersist, whereas if they decay inanoscillatory fashion , then multiple regions can persist

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Multiple bumps in a neuronal model of working memory

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Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopft bifurcation

The bistable Swift-Hohenberg equation exhibits multiple stable and unstable spatially localized states of arbitrary length in the vicinity of the Maxwell point between spatially homogeneous and periodic states. These states are organized in a characteristic snakes-and-ladders structure. The origin of this structure in one spatial dimension is reviewed, and the stability properties of the resulting states with respect to perturbations in both one and two dimensions are described. The relevance of the results to several different physical systems is discussed.

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Homoclinic snaking: structure and stability.

A long structural system with an unstable (subcritical)post-buckling response that subsequently restabilizes typically deformsin a cellular manner, with localized buckles first forming and thenlocking up in sequence. As buckling continues over a growing number ofcells, the response can be described by a set of lengthening homoclinicconnections from the fundamental equilibrium state to itself. In thelimit, this leads to a heteroclinic connection from the fundamentalunbuckled state to a post-buckled state that is periodic. Under suchprogressive displacement the load tends to oscillate between twodistinct values. The paper is both a review and a pointer tofuture research. The response is described via a typical system, asimple but ubiquitous model of a strut on a foundation which includesinitially-destabilizing and finally-restabilizing nonlinear terms. Anumber of different structural forms, including the axially-compressedcylindrical shell, a typical sandwich structure, a model of geologicalfolding and a simple link model are shown to display such behaviour. Amathematical variational argument is outlined for determining the globalminimum postbuckling state under controlled end displacement (rigidloading). Finally, the paper stresses the practical significance of aMaxwell-load instability criterion for such systems. This criterion,defined under dead loading to be where the pre-buckled and post-buckledstate have the same energy, is shown to have significance in the presentsetting under rigid loading also. Specifically, the Maxwell load isargued to be the limit of minimum energy localized solutions asend-shortening tends to infinity.

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Cellular Buckling in Long Structures

Compressed sandwich structures, comprising two stiff face plates separated by a softer core material, while designed principally as efficient integral structures, can lose this quality when faces buckle locally. Interaction between overall (Euler) buckling and local buckling of one face suggests that failure will localize into the centre. A variational formulation, leading to a pair of nonlinear differential equations subject to integral constraints, describes the post–buckling response. These are solved by a combination of numerical shooting and continuation techniques, such that the response far into the unstable post–buckling regime can be portrayed. Solutions with both linear and nonlinear constitutive core relations are compared with the results of an engineering (body–force) approach, and with those of earlier (periodic) Rayleigh–Ritz analyses. The latter demonstrate the extra destabilization that comes with localization.

Localization and mode interaction in sandwich structures

Length scale interactions in the folding of sandwich structures

Cellular buckling in long structures. The theme of solitary waves and localization phenomena in elastic structures

The phenomenon on kink banding, occurring primarily in layered structures under layer-parallel compression, has been studied extensively in a variety of different situations. Examples of physical systems which exhibit this phenomenon include the deformation of geological strata [1, 2], compressed laminated fibre composites [3, 4, 5], and internally in fibre and wire ropes [6]. In our earlier work, the focus was on the geological application of kink banding where layers are confined by pressure and therefore penalize the formation of voids in the structure. Some of the technical difficulties that are encountered in the modelling of such systems were outlined in terms of using continuum mechanics and Cosserat-type continua. The approach we adopted involved formulating a model that cut out most of the complexity, but kept large rotations—the nonlinearity that seems to govern the structural response. This simple model exhibits a path of pure squash and a path of deformation where the kink bands form and rotate. However, as the paths of equilibrium cross at infinity, i.e. there exists an infinite critical load, from the linearized viewpoint there is no such instability, a conclusion that is not confirmed by experimental evidence which clearly shows the structure deforming suddenly after a certain amount of loading. The Maxwell stability criterion [7] was used in an earlier paper to overcome this difficulty [8]. This allowed the evaluation of a robust lower bound of the displacement where the system jumped from the undeformed to the kinked state, and the evaluation of the lock-up angle of the kink band. In the current study, several enhancements of the original model are outlined such that quantitative comparisons can be made between experiments and theory away from the neighbourhood of the initial instability and significantly into the post-kinking regime. It is found that not only can the new model embrace mechanisms for band propagation, in terms of band broadening and progression, but that comparisons of the band width and orientation against experiments are excellent. 2 Model Characteristics

M.A. Wadee

Papers

Cellular Buckling in Long Structures

Localization and mode interaction in sandwich structures

Length scale interactions in the folding of sandwich structures

Cellular buckling in long structures. The theme of solitary waves and localization phenomena in elastic structures

Simplified kink banding: a predictive model for localized failure in layered structures