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We investigate stationary, spatially localized patterns in lattice dynamical systems that exhibit bistability. The profiles associated with these patterns have a long plateau where the pattern resembles one of the bistable states, while the profile is close to the second bistable state outside this plateau. We show that the existence branches of such patterns generically form either an infinite stack of closed loops (isolas) or intertwined s-shaped curves (snaking). We then use bifurcation theory near the anti-continuum limit, where the coupling between edges in the lattice vanishes, to prove existence of isolas and snaking in a bistable discrete real Ginzburg–Landau equation. We also provide numerical evidence for the existence of snaking diagrams for planar localized patches on square and hexagonal lattices and outline a strategy to analyse them rigorously. [ABSTRACT FROM AUTHOR]

Abstract Spatially localized patterns have been observed in numerous physical contexts, and their bifurcation diagrams often exhibit similar snaking behavior: symmetric solution branches, connected by bifurcating asymmetric solution branches, wind back and forth in an appropriate parameter. Previous papers have addressed existence of such solutions; here we address their stability, taking the necessary first step of unifying existence and uniqueness proofs for symmetric and asymmetric solutions. We then show that, under appropriate assumptions, temporal eigenvalues of the front and back underlying a localized solution are added with multiplicity in the right half plane. In a companion paper, we analyze the behavior of eigenvalues at λ = 0 and inside the essential spectrum. Our results show that localized snaking solutions are stable if, and only if, the underlying fronts and backs are stable: unlike localized non-oscillatory solutions, no interaction eigenvalues are present. We use the planar Swift–Hohenberg system to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we analyse bifurcations from stationary stable spots to travelling spots in a planar three-component FitzHugh–Nagumo system that was proposed previously as a phenomenological model of gas-discharge systems. By combining formal analyses, centre-manifold reductions, and detailed numerical continuation studies, we show that, in the parameter regime under consideration, the stationary spot destabilizes either through its zeroth Fourier mode in a Hopf bifurcation or through its first Fourier mode in a pitchfork or drift bifurcation, whilst the remaining Fourier modes appear to create only secondary bifurcations. Pitchfork bifurcations result in travelling spots, and we derive criteria for the criticality of these bifurcations. Our main finding is that supercritical drift bifurcations, leading to stable travelling spots, arise in this model, which does not seem possible for its two-component version. [Copyright &y& Elsevier]

Abstract: We expand upon a general framework for studying the bifurcation diagrams of localized spatially oscillatory structures. Building on work by Beck et al., the present work provides rigorous analytical results on the effects of perturbations to systems exhibiting snaking behavior. Starting with a reversible variational system possessing an additional symmetry, we elucidate the distinct effects of breaking symmetry and breaking variational structure, and characterize the resulting changes in both the bifurcation diagram and the solutions themselves. We show how to predict the branch reorganization and drift speeds induced by any particular given perturbative term, and illustrate our results via numerical continuation. We further demonstrate the utility of our methods in understanding the effects of particular perturbations breaking reversibility. Our approach yields an analytical explanation for previous numerical results on the effects of perturbations in the one-dimensional cubic–quintic Swift–Hohenberg model and allows us to make predictions on the effects of perturbations in more general settings, including planar systems. While our numerical results involve the Swift–Hohenberg model system, we emphasize the general applicability of the analytical results. [Copyright &y& Elsevier]

Localized planar patterns arise in many reaction-diffusion models. Most of the paradigm equations that have been studied so far are two-component models. While stationary localized structures are often found to be stable in such systems, travelling patterns either do not exist or are found to be unstable. In contrast, numerical simulations indicate that localized travelling structures can be stable in three-component systems. As a first step towards explaining this phenomenon, a planar singularly perturbed three-component reaction-diffusion system that arises in the context of gas-discharge systems is analysed in this paper. Using geometric singular perturbation theory, the existence and stability regions of radially symmetric stationary spot solutions are delineated and, in particular, stable spots are shown to exist in appropriate parameter regimes. This result opens up the possibility of identifying and analysing drift and Hopf bifurcations, and their criticality, from the stationary spots described here. [ABSTRACT FROM AUTHOR]

Abstract: The bifurcation structure of localized stationary radial patterns of the Swift–Hohenberg equation is explored when a continuous parameter is varied that corresponds to the underlying space dimension whenever is an integer. In particular, we investigate how 1D pulses and 2-pulses are connected to planar spots and rings when is increased from 1 to 2. We also elucidate changes in the snaking diagrams of spots when the dimension is switched from 2 to 3. [Copyright &y& Elsevier]

We investigate stationary spatially localized hexagon patterns of the two-dimensional (2D) Swift- Hohenberg equation in the parameter region where the trivial state and regular hexagon patterns are both stable. Using numerical continuation techniques, we trace out the existence regions of fully localized hexagon patches and of planar pulses which consist of a strip filled with hexagons that is embedded in the trivial state. We find that these patterns exhibit snaking: for each parameter value in the snaking region, an infinite number of patterns exist that are connected in parameter space and whose width increases without bound. Our computations also indicate a relation between the limits of the snaking regions of planar hexagon pulses with different orientations and of the fully localized hexagon patches. To investigate which hexagons among the one-parameter family of hexagons are selected in a hexagon pulse or front, we derive a conserved quantity of the spatial dynamical system that describes planar patterns which are periodic in the transverse direction and use it to calculate the Maxwell curves along which the selected hexagons have the same energy as the trivial state. We find that the Maxwell curve lies within the snaking region, as expected from heuristic arguments. [ABSTRACT FROM AUTHOR]

Motivated by experimental observations in the light-sensitive Belousov—Zhabotinsky reaction and subsequent numerical works, we discuss period-doubling bifurcations of spiral waves and other coherent structures. We report on explanations of the observed phenomena which involve a detailed analysis of spectra, and of the associated eigenfunctions, of defects on bounded and unbounded domains. [ABSTRACT FROM AUTHOR]

Fronts are traveling waves in spatially extended systems that connect two different spatially homogeneous rest states. If the rest state behind the front undergoes a supercritical Turing instability, then the front will also destabilize. On the linear level, however, the front will be only convectively unstable since perturbations will be pushed away from the front as it propagates. In other words, perturbations may grow, but they can do so only behind the front. The goal of this paper is to prove for a specific model system that this behavior carries over to the full nonlinear system. [ABSTRACT FROM AUTHOR]

In this article, existence and stability of N-front travelling-wave solutions of partial differential equations on the real line is investigated. The N-fronts considered here arise as heteroclinic orbits bifurcating from a twisted heteroclinic loop in the underlying ordinary differential equation describing travelling-wave solutions. It is proved that the N-front solutions are linearly stable provided the fronts building the twisted heteroclinic loop are linearly stable. The result is applied to travelling waves arising in the FitzHugh-Nagumo equation. [ABSTRACT FROM AUTHOR]

Abstract: Reaction–diffusion equations on the real line that contain a control parameter are investigated. Of interest are travelling front solutions for which the rest state behind the front undergoes a supercritical Turing or Hopf bifurcation as the parameter is increased. This causes the essential spectrum to cross into the right half plane, leading to a linear convective instability in which the emerging pattern is pushed away from the front as it propagates. It is shown, however, that the wave remains nonlinearly stable in an appropriate sense. More precisely, using the fact that the instability is supercritical, it is shown that the amplitude of any pattern that emerges behind the wave saturates at some small parameter-dependent level and that the pattern is pushed away from the front interface. As a result, when considered in an appropriate exponentially weighted space, the travelling front remains stable, with an exponential in time rate of convergence. [Copyright &y& Elsevier]

Invariant manifolds are key objects in describing how trajectories partition the phase spaces of a dynamical system. Examples include stable, unstable, and center manifolds of equilibria and periodic orbits, quasiperiodic invariant tori, and slow manifolds of systems with multiple timescales. Changes in these objects and their intersections with variation of system parameters give rise to global bifurcations. Bifurcation manifolds in the parameter spaces of multi-parameter families of dynamical systems also play a prominent role in dynamical systems theory. Much progress has been made in developing theory and computational methods for invariant manifolds during the past 25 years. This article highlights some of these achievements and remaining open problems. [ABSTRACT FROM AUTHOR]

The optimal-velocity follow-the-leader model is augmented with an equation that allows each driver to adjust their target headway according to the velocity difference between the driver and the car in front. In this more detailed model, which is investigated on a ring, stable and unstable multipulse or multijam solutions emerge. Analytical investigations using truncated Fourier analysis are confirmed and complemented by a detailed numerical bifurcation analysis. In addition to standard rotating waves, time-modulated waves are found. [ABSTRACT FROM AUTHOR]

Stable localized roll structures have been observed in many physical problems and model equations, notably in the one-dimensional (1D) Swift--Hohenberg equation. Reflection-symmetric localized rolls are often found to lie on two "snaking" solution branches so that the spatial width of the localized rolls increases when moving along each branch. Recent numerical results by Burke and Knobloch indicate that the two branches are connected by infinitely many "ladder" branches of asymmetric localized rolls. In this paper, these phenomena are investigated analytically. It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls. It is also shown that isolas of asymmetric states may exist, and it is argued that the results presented here apply to 2D stationary states that are localized in one spatial direction. [ABSTRACT FROM AUTHOR]