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![An illustration of the unfolding of the pitchfork bifurcation for d≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ne 0$$\end{document}](publication/335703294/figure/fig2/AS:960073979396096@1605910882075/An-illustration-of-the-unfolding-of-the-pitchfork-bifurcation-for.png)
An illustration of the unfolding of the pitchfork bifurcation for d≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ne 0$$\end{document}
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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 su...
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Citations
... The correspondence between the continuous and discrete settings can be furthered since the map (2.23) can be used as an example of a Poincaré map near the periodic orbits of (2.10) that exhibits a homoclinic tangle and leads to the plethora of homoclinic orbits that correspond to localized solutions. Bramburger and Sandstede [45] used the (discrete) spatial dynamics formulation (2.23) to provide a complete explanation of the snaking structure of localized solutions, with [42] extending these results to prove that multipulses must always lie along isolas. Most importantly though, these studies employed perturbation theory and Lyapunov-Schmidt reductions to completely identify the bifurcation of front/back solutions of (2.22) when 0 < d ≪ 1. ...
... Most importantly though, these studies employed perturbation theory and Lyapunov-Schmidt reductions to completely identify the bifurcation of front/back solutions of (2.22) when 0 < d ≪ 1. These results therefore provide a method of explicitly checking all hypotheses required to obtain the results of [42,45], thus bridging the gap from the theory to application in specific systems. ...
Localized patterns are coherent structures embedded in a quiescent state and occur in both discrete and continuous media across a wide range of applications. While it is well-understood how domain covering patterns (for example stripes and hexagons) emerge from a pattern-forming/Turing instability, analyzing the emergence of their localized counterparts remains a significant challenge. There has been considerable progress in studying localized patterns over the past few decades, often by employing innovative mathematical tools and techniques. In particular, the study of localized pattern formation has benefited greatly from numerical techniques; the continuing advancement in computational power has helped to both identify new types patterns and further our understanding of their behavior. We review recent advances regarding the complex behavior of localized patterns and the mathematical tools that have been developed to understand them, covering various topics from spatial dynamics, exponential asymptotics, and numerical methods. We observe that the mathematical understanding of localized patterns decreases as the spatial dimension increases, thus providing significant open problems that will form the basis for future investigations.
... However, our proofs are constructive in that they explicitly demonstrate that the long trajectories are built up by shadowing the 'base' homoclinic and heteroclinic orbits found to exist by the numerical methods herein. This comes from the fact that our proofs are similar to the work in [7,10], whereby we move to local Poincaré sections near the UPOs and use the existence of heteroclinic connections to transit between these local sections. The result is a general description of saddle mediated transport in two degree of freedom Hamiltonian systems that goes beyond the existing theory. ...
... Precisely, the saddle structure gives that orbits enter the neighborhood closely following the stable manifold and exit the neighborhood closely following the unstable manifold, all while wrapping around the periodic orbit. This is made precise with the following lemma which originally appeared in [10] and comes as the discretetime analogue of the main result of [69]. for all n ∈ {0, . . . ...
Pendulums are simple mechanical systems that have been studied for centuries and exhibit many aspects of modern dynamical systems theory. In particular, the double pendulum is a prototypical chaotic system that is frequently used to illustrate a variety of phenomena in nonlinear dynamics. In this work, we explore the existence and implications of codimension-1 invariant manifolds in the double pendulum, which originate from unstable periodic orbits around saddle equilibria and act as separatrices that mediate the global phase space transport. Motivated in part by similar studies on the three-body problem, we are able to draw a direct comparison between the dynamics of the double pendulum and transport in the solar system, which exist on vastly different scales. Thus, the double pendulum may be viewed as a table-top benchmark for chaotic, saddle-mediated transport, with direct relevance to energy-efficient space mission design. The analytical results of this work provide an existence result, concerning arbitrarily long itineraries in phase space, that is applicable to a wide class of two degree of freedom Hamiltonian systems, including the three-body problem and the double pendulum. This manuscript details a variety of periodic orbits corresponding to acrobatic motions of the double pendulum that can be identified and controlled in a laboratory setting.
... However, our proofs are constructive in that they explicitly demonstrate that the long trajectories are built up by shadowing the 'base' homoclinic and heteroclinic orbits found to exist by the numerical methods herein. This comes from the fact that our proofs are similar to the work in [48,49], whereby we move to local Poincaré sections near the UPOs and use the existence of heteroclinic connections to transit between these local sections. The result is a general description of saddle mediated transport in two degree of freedom Hamiltonian systems that goes beyond the existing theory. ...
... Precisely, the saddle structure gives that orbits enter the neighborhood closely following the stable manifold and exit the neighborhood closely following the unstable manifold, all while wrapping around the periodic orbit. This is made precise with the following lemma which originally appeared in [48] and comes as the discretetime analogue of the main result of [67]. for all n ∈ {0, . . . ...
Pendulums are simple mechanical systems that have been studied for centuries and exhibit many aspects of modern dynamical systems theory. In particular, the double pendulum is a prototypical chaotic system that is frequently used to illustrate a variety of phenomena in nonlinear dynamics. In this work, we explore the existence and implications of codimension-1 invariant manifolds in the double pendulum, which originate from unstable periodic orbits around saddle equilibria and act as separatrices that mediate the global phase space transport. Motivated in part by similar studies on the three-body problem, we are able to draw a direct comparison between the dynamics of the double pendulum and transport in the solar system, which exist on vastly different scales. Thus, the double pendulum may be viewed as a table-top benchmark for chaotic, saddle-mediated transport, with direct relevance to energy-efficient space mission design. The analytical results of this work provide an existence result, concerning arbitrarily long itineraries in phase space, that is applicable to a wide class of two degree of freedom Hamiltonian systems, including the three-body problem and the double pendulum. This manuscript details a variety of periodic orbits corresponding to acrobatic motions of the double pendulum that can be identified and controlled in a laboratory setting.
... For lattices, results about pinning of onedimensional and planar fronts near the continuum limit were obtained in [25,26], respectively, using asymptotics-beyond-all-orders methods. Results about snaking diagrams of localized patterns in one-dimensional lattices were recently established in [27]. ...
... As mentioned earlier, not much is known analytically for localized planar patterns, and one of the reasons is that the techniques used in the one-dimensional case rely primarily on formulating the existence problem as a spatial dynamical system in the unbounded spatial variable, so that localized structures can be viewed and constructed as homoclinic orbits [12,27,34]. This approach is no longer available for genuinely planar patterns. ...
... Since our analysis in the anti-continuum limit relied on Lyapunov-Schmidt reduction, these cases should be amenable to analysis as well, and it would be interesting to see whether the bifurcation diagrams are similar. Finally, other lattices could be explored: we refer to [27,33] for numerical computations of localized patterns on hexagonal lattices, to [33] for triangular lattices, and to [37] for a numerical study of snaking of localized patterns in a predator-prey model on Barabási-Albert networks, where the coupling operator is given by the graph Laplacian. ...
Localized planar patterns in spatially extended bistable systems are known to exist along intricate bifurcation diagrams, which are commonly referred to as snaking curves. Their analysis is challenging as techniques such as spatial dynamics that have been used to explain snaking in one space dimension no longer work in the planar case. Here, we consider bistable systems posed on square lattices and provide an analytical explanation of snaking near the anti-continuum limit using Lyapunov--Schmidt reduction. We also establish stability results for localized patterns, discuss bifurcations to asymmetric states, and provide further numerical evidence that the shape of snaking curves changes drastically as the coefficient that reflects the strength of the spatial coupling crosses a finite threshold.
... The main difference between the diagrams when a = 21 and a = 32 is that in the latter the slice through the two-parameter bifurcation diagram crosses the cyan curve, at b = 96.5081. The crossing with the cyan curve leads to the appearance of a stack of isolas [9,13,18,37]. Their formation may be associated with multipulse orbits that exist independently of the snaking bifurcation, as in [12], or due to the so called defect mediated solutions [18,35]. In Fig. 3(A) we show five isolas, these are not connected to the snaking bifurcations. ...
The Thomas model is a system of two reaction–diffusion equations which was originally proposed in the context of enzyme kinetics. It was subsequently realised that it offers a plausible chemical mechanism for the generation of coat markings on mammals. To that end previous investigations have focused on establishing the conditions for the Turing instability and on following the associated patterns as the bifurcation parameter increases through the instability.
In this paper we use modern ideas from the theory of dynamical systems to systematically investigate the formation of localised structures in this model. The Turing instability is found to exhibit a degeneracy at which it changes from being subcritical to supercritical (or vice versa). Associated with such degenerate points is a heteroclinic connection which is a crucial requirement for the generation of localised patterns.
Localised solutions containing a single ‘spike’ are associated with the so-called Belyakov-Devaney transition. Localised solutions containing either multiple ‘peaks’ or multiple ‘holes’ are associated with homoclinic snaking bifurcations. These structures are investigated using continuation software. The snaking bifurcations are found to collapse when the system crosses the Belyakov-Devaney transition. The isolated spike solutions collapse at a codimension two heteroclinic connection to create the so-called collapsed snaking.
We show that the temporal stability of the localised solutions depends upon the presence of Hopf bifurcations which in turn are controlled by the diffusion ratio. For small values of the ratio only solutions in the spike region are destabilised by a Hopf bifurcation. For larger values of the diffusion ratio Hopf bifurcations destabilise most of the localised patterns.
... We also mention recent work on Turing patterns in zebrafish resulting from nonlocal, distance-dependent interactions between cells [24,42]. In a somewhat different direction, the study of localized pattern formation in the discrete spatial setting has been studied previously on infinite chains [5,6], square lattices [7], and rings [40]. We also mention in passing the prevalence of non-local, distance dependent coupling in neural field models; see for example [1,8,18]. ...
... This can be observed, for example, in Figure 2 for the small-world network. The competition between states in regions of bistability has been shown to result in steady-state localized patterns for which a connected subset of the elements are near the stable patterned state, while the remaining elements on the graph are near the homogeneous state u = 0 [6,7,40]. It remains to identify if such localized states exist in the small regions of bistability observed in the random graph networks studied here. ...
We study Turing bifurcations on one-dimensional random ring networks where the probability of a connection between two nodes depends on the distance between the two nodes. Our approach uses the theory of graphons to approximate the graph Laplacian in the limit as the number of nodes tends to infinity by a nonlocal operator -- the graphon Laplacian. For the ring networks considered here, we employ center manifold theory to characterize Turing bifurcations in the continuum limit in a manner similar to the classical partial differential equation case and classify these bifurcations as sub/super/trans-critical. We derive estimates that relate the eigenvalues and eigenvectors of the finite graph Laplacian to those of the graphon Laplacian. We are then able to show that, for a sufficiently large realization of the network, with high probability the bifurcations that occur in the finite graph are well approximated by those in the graphon limit. The number of nodes required depends on the spectral gap between the critical eigenvalue and the remaining ones, with the smaller this gap the more nodes that are required to guarantee that the graphon and graph bifurcations are similar. We demonstrate that if this condition is not satisfied then the bifurcations that occur in the finite network can differ significantly from those in the graphon limit.
... Many of these investigations were motivated by the observation of snaking patterns in experiments and models, including in ferrofluids [18,24,30], optical systems [9,15,17], and vegetation models [27], to name but a few. Similarly complex bifurcation structures of localized solutions have also been observed in spatially discrete systems posed on integer lattices in [9,10,22,23,26,28,32,34,35] and were explained in part by [3][4][5]. While the latter works have explained the bifurcation structure of localized solutions on "regular" graphs, such as the integer lattices, little is known about how graph structure and connection topology influence the connections of localized solutions in parameter space. ...
We study the structure of stationary patterns in bistable lattice dynamical systems posed on rings with a symmetric coupling structure in the regime of small coupling strength. We show that sparse coupling (for instance, nearest-neighbour or next-nearest-neighbour coupling) and all-to-all coupling lead to significantly different solution branches. In particular, sparse coupling leads to snaking branches with many saddle-node bifurcations, whilst all-to-all coupling leads to branches with six saddle nodes, regardless of the size of the number of nodes in the graph.
... As with the single pulses, each of the multi-pulses will be stable when 1 = 0, and they will persist as stable structures for sufficiently small 1 . Typically, the construction of multi-pulses would involve a discussion of tail-tail interactions between individual pulses, and an application of the Hale-Lin-Sandstede method (e.g., see [5,6,16,24,[26][27][28][29] and the references therein). However, for the system under consideration this is less relevant, as the nonlinear coupling between adjacent sites renders the transition from one state to another to be super-exponential, instead of the exponential rates associated with linear coupling (see Figure 6 for a representative demonstration of this phenomena). ...
A language dynamics model on a square lattice, which is an extension of the one popularized by Abrams and Strogatz [1], is analyzed using ODE bifurcation theory. For this model we are interested in the existence and spectral stability of structures such as stripes, which are realized through pulses and/or the concatenation of fronts, and spots, which are a contiguous collection of sites in which one language is dominant. Because the coupling between sites is nonlinear, the boundary between sites containing speaking two different languages is "sharp"; in particular, in a PDE approximation it allows for the existence of compactly supported pulses (compactons). The dynamics are considered as a function of the prestige of a language. In particular, it is seen that as the prestige varies, it allows for a language to spread through the network, or conversely for its demise.
... Furthermore, symmetric single-pulses of (1.1) lie along unbounded curves that bounce back and forth between fixed values of µ, while the length of the region of activation monotonically increases without bound. Such a bifurcation scenario is termed snaking and it has been documented extensively in lattice dynamical systems [2,10,11,23,26,30,35,37,42]. Beyond the symmetric single-pulses, there also exist asymmetric singlepulses which bifurcate from the symmetric snaking branches in a pitchfork bifurcation. ...
... In particular, it has been shown that the specific form of the bifurcation curves of single-pulse solutions are entirely dictated by the bifurcation structure of front solutions which asymptotically connect the homogeneous background state to the patterned or activated state. Recently these results were extended to lattice dynamical systems in [2] and fully explain the organization of bifurcation curves in Figure 1. Furthermore, in the case of (1.1) we can exploit the anticontinuum limit, corresponding to the uncoupled system arising when setting d = 0, to verify the conditions of the general theory for 0 < d 1. ...
... Following [2], steady-state solutions of (1.1) are bounded solutions of the discrete dynamical system ...
This work investigates the existence and bifurcation structure of multi-pulse steady-state solutions to bistable lattice dynamical systems. Such solutions are characterized by multiple compact disconnected regions where the solution resembles one of the bistable states and resembles another trivial bistable state outside of these compact sets. It is shown that the bifurcation curves of these multi-pulse solutions lie along closed and bounded curves (isolas), even when single-pulse solutions lie along unbounded curves. These results are applied to a discrete Nagumo differential equation and we show that the hypotheses of this work can be confirmed analytically near the anti-continuum limit. Results are demonstrated with a number of numerical investigations.
... For lattices, results about pinning of onedimensional and planar fronts near the continuum limit were obtained in [25,26], respectively, using asymptotics-beyond-all-orders methods. Results about snaking diagrams of localized patterns in one-dimensional lattices were recently established in [27]. ...
... As mentioned earlier, not much is known analytically for localized planar patterns, and one of the reasons is that the techniques used in the one-dimensional case rely primarily on formulating the existence problem as a spatial dynamical system in the unbounded spatial variable, so that localized structures can be viewed and constructed as homoclinic orbits [12,27,34]. This approach is no longer available for genuinely planar patterns. ...
... Since our analysis in the anti-continuum limit relied on Lyapunov-Schmidt reduction, these cases should be amenable to analysis as well, and it would be interesting to see whether the bifurcation diagrams are similar. Finally, other lattices could be explored: we refer to [27,33] for numerical computations of localized patterns on hexagonal lattices, to [33] for triangular lattices, and to [37] for a numerical study of snaking of localized patterns in a predator-prey model on Barabási-Albert networks, where the coupling operator is given by the graph Laplacian. ...
Localized planar patterns in spatially extended bistable systems are known to exist along intricate bifurcation diagrams, which are commonly referred to as snaking curves. Their analysis is challenging as techniques such as spatial dynamics that have been used to explain snaking in one space dimension no longer work in the planar case. Here, we consider bistable systems posed on square lattices and provide an analytical explanation of snaking near the anti-continuum limit using Lyapunov–Schmidt reduction. We also establish stability results for localized patterns, discuss bifurcations to asymmetric states, and provide further numerical evidence that the shape of snaking curves changes drastically as the coefficient that reflects the strength of the spatial coupling crosses a finite threshold.
