Localised states in an extended Swift-Hohenberg equation

Recent work on the behaviour of localised states in pattern forming partial differential equations has focused on the traditional model Swift-Hohenberg equation which, as a result of its simplicity, has additional structure --- it is variational in time and conservative in space. In this paper we investigate an extended Swift-Hohenberg equation in which non-variational and non-conservative effects play a key role. Our work concentrates on aspects of this much more complicated problem. Firstly we carry out the normal form analysis of the initial pattern forming instability that leads to small-amplitude localised states. Next we examine the bifurcation structure of the large-amplitude localised states. Finally we investigate the temporal stability of one-peak localised states. Throughout, we compare the localised states in the extended Swift-Hohenberg equation with the analogous solutions to the usual Swift-Hohenberg equation

The Swift-Hohenberg equation with a nonlocal nonlinearity

It is well known that aspects of the formation of localised states in a one-dimensional Swift--Hohenberg equation can be described by Ginzburg--Landau-type envelope equations. This paper extends these multiple scales analyses to cases where an additional nonlinear integral term, in the form of a convolution, is present. The presence of a kernel function introduces a new lengthscale into the problem, and this results in additional complexity in both the derivation of envelope equations and in the bifurcation structure. When the kernel is short-range, weakly nonlinear analysis results in envelope equations of standard type but whose coefficients are modified in complicated ways by the nonlinear nonlocal term. Nevertheless, these computations can be formulated quite generally in terms of properties of the Fourier transform of the kernel function. When the lengthscale associated with the kernel is longer, our method leads naturally to the derivation of two different, novel, envelope equations that describe aspects of the dynamics in these new regimes. The first of these contains additional bifurcations, and unexpected loops in the bifurcation diagram. The second of these captures the stretched-out nature of the homoclinic snaking curves that arises due to the nonlocal term.Comment: 28 pages, 14 figures. To appear in Physica

Resonance bifurcations from robust homoclinic cycles

We present two calculations for a class of robust homoclinic cycles with symmetry Z_n x Z_2^n, for which the sufficient conditions for asymptotic stability given by Krupa and Melbourne are not optimal. Firstly, we compute optimal conditions for asymptotic stability using transition matrix techniques which make explicit use of the geometry of the group action. Secondly, through an explicit computation of the global parts of the Poincare map near the cycle we show that, generically, the resonance bifurcations from the cycles are supercritical: a unique branch of asymptotically stable period orbits emerges from the resonance bifurcation and exists for coefficient values where the cycle has lost stability. This calculation is the first to explicitly compute the criticality of a resonance bifurcation, and answers a conjecture of Field and Swift in a particular limiting case. Moreover, we are able to obtain an asymptotically-correct analytic expression for the period of the bifurcating orbit, with no adjustable parameters, which has not proved possible previously. We show that the asymptotic analysis compares very favourably with numerical results.Comment: 24 pages, 3 figures, submitted to Nonlinearit

After 1952:the later development of Alan Turing's ideas on the mathematics of pattern formation

The paper 'The chemical basis of morphogenesis' [. Phil. Trans. R. Soc. Lond. B 237, 37-72 (1952)] by Alan Turing remains hugely influential in the development of mathematical biology as a field of research and was his only published work in the area. In this paper I discuss the later development of his ideas as revealed by lesser-known archive material, in particular the draft notes for a paper with the title 'Outline of development of the Daisy'.These notes show that, in his mathematical work on pattern formation, Turing developed substantial insights that go far beyond Turing (1952). The model differential equations discussed in his notes are substantially different from those that are the subject of Turing (1952) and present a much more complex mathematical challenge. In taking on this challenge, Turing's work anticipates (i) the description of patterns in terms of modes in Fourier space and their nonlinear interactions, (ii) the construction of the well-known model equation usually ascribed to Swift and Hohenberg, published 23 years after Turing's death, and (iii) the use of symmetry to organise computations of the stability of symmetrical equilibria corresponding to spatial patterns.This paper focuses on Turing's mathematics rather than his intended applications of his theories to phyllotaxis, gastrulation, or the unicellular marine organisms Radiolaria. The paper argues that this archive material shows that Turing encountered and wrestled with many issues that became key mathematical research questions in subsequent decades, showing a level of technical skill that was clearly both ahead of contemporary work, and also independent of it. His legacy in recognising that the formation of patterns can be understood through mathematical models, and that this mathematics could have wide application, could have been far greater than just the single paper of 1952.A revised and substantially extended draft of 'Outline of development of the Daisy' is included in the Supplementary material. L'article unique et célèbre d'Alan Turing 'The chemical basis of morphogenesis' [. Phil. Trans. R. Soc. Lond. B 237, 37-72 (1952)] reste encore aujourd'hui très influent dans l'essor de la biologie mathématique. Ici, je discute les développements ultérieurs des idées de Turing révélées par des documents d'archives moins connus, en particulier son projet d'article intitulé 'Outline of development of the Daisy'.Ces documents, replacés dans l'oeuvre mathématique de Turing sur la morphogénèse et la formation de motifs, témoignent d'avancées majeures qui vont bien au-delà de l'article de 1952. En effet, Turing aborde dans ses notes des équations différentielles sensiblement différentes de celles de 1952 qui constituent un problème de mathématique d'un abord beaucoup plus complexe.Embrassant ce défi, Turing propose (i) une description des motifs réguliers sous la forme de modes de Fourier et de leurs intéractions non-linéaires, (ii) la construction de l'équation modèle bien connue de Swift et Hohenberg, publiée 23 ans après la mort de Turing, et enfin (iii) l'utilisation des propriétés de symétrie de ces équations d'évolution afin d'organiser et de simplifier les calculs nécessaires à l'étude de stabilité des équilibres symétriques correspondant aux motifs spatiaux.Dans cet article, l'accent est porté sur les mathématiques de Turing et non sur les applications de ses théories à la phyllotaxie, la gastrulation, ou encore sur la morphogénèse des organismes marins unicellulaires comme les Radiolaria. On y montre en particulier que Turing s'est confronté à de nombreux problèmes ardus qui sont devenus dans les décennies suivantes des questions majeures en recherche mathématique, ce qui démontre une fois de plus un niveau de compétence technique hors norme qui était clairement à la fois bien en avance sur son temps, mais aussi indépendant de celui-ci. En reconnaissant que la formation de motifs peut se comprendre grâce à des modèles mathématiques, aux vastes champs d'application, il est évident que l'héritage de Turing aurait pu être beaucoup plus important que celui de son papier de 1952.Une reproduction sensiblement révisée et complétée de son ébauche d'article 'Outline of development of the Daisy' est incluse en annexe.</p

Dynamics near a periodically forced robust heteroclinic cycle

In this article we discuss, with a combination of analytical and numerical results, a canonical set of differential equations with a robust heteroclinic cycle, subjected to time-periodic forcing. We find that three distinct dynamical regimes exist, depending on the ratio of the contracting and expanding eigenvalues at the equilibria on the heteroclinic cycle which exists in the absence of forcing. By reducing the dynamics to that of a two dimensional map we show how frequency locking and complex dynamics arise