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Echo State Networks (ESNs) are a class of single layer recurrent neural networks that have enjoyed recent attention. In this paper we prove that a suitable ESN, trained on a series of measurements of an invertible dynamical system, induces a C1 map from the dynamical system's phase space to the ESN's reservoir space. We call this the Echo State Map. We then prove that the Echo State Map is generically an embedding with positive probability. Under additional mild assumptions, we further conjecture that the Echo State Map is almost surely an embedding. For sufficiently large, and specially structured, but still randomly generated ESNs, we prove that there exists a linear readout layer that allows the ESN to predict the next observation of a dynamical system arbitrarily well. Consequently, if the dynamical system under observation is structurally stable then the trained ESN will exhibit dynamics that are topologically conjugate to the future behaviour of the observed dynamical system. Our theoretical results connect the theory of ESNs to the delay-embedding literature for dynamical systems, and are supported by numerical evidence from simulations of the traditional Lorenz equations. The simulations confirm that, from a one dimensional observation function, an ESN can accurately infer a range of geometric and topological features of the dynamics such as the eigenvalues of equilibrium points, Lyapunov exponents and homology groups., Comment: 24 pages, 9 figures

Neural crest cells are crucial in development, not least because of their remarkable multipotency. Early findings stimulated two hypotheses for how fate specification and commitment from fully multipotent neural crest cells might occur, progressive fate restriction (PFR) and direct fate restriction, differing in whether partially restricted intermediates were involved. Initially hotly debated, they remain unreconciled, although PFR has become favoured. However, testing of a PFR hypothesis of zebrafish pigment cell development refutes this view. We propose a novel ‘cyclical fate restriction’ hypothesis, based upon a more dynamic view of transcriptional states, reconciling the experimental evidence underpinning the traditional hypotheses.

Understanding cell fate selection remains a central challenge in developmental biology. We present a class of simple yet biologically-motivated mathematical models for cell differentiation that generically generate oscillations and hence suggest alternatives to the standard framework based on Waddington’s epigenetic landscape. The models allow us to suggest two generic dynamical scenarios that describe the differentiation process. In the first scenario gradual variation of a single control parameter is responsible for both entering and exiting the oscillatory regime. In the second scenario two control parameters vary: one responsible for entering, and the other for exiting the oscillatory regime. We analyse the standard repressilator and four variants of it and show the dynamical behaviours associated with each scenario. We present a thorough analysis of the associated bifurcations and argue that gene regulatory networks with these repressilator-like characteristics are promising candidates to describe cell fate selection through an oscillatory process.

Neural crest cells (NCCs) are highly multipotent stem cells. A long-standing controversy exists over the mechanism of NCC fate specification, specifically regarding the presence and potency of intermediate progenitors. The direct fate restriction (DFR) model, based on early in vivo clonal studies, hypothesised that intermediates are absent and that migrating cells maintain full multipotency1–6. However, most authors favour progressive fate restriction (PFR) models, with fully multipotent early NCCs (ENCCs) transitioning to partially-restricted intermediates before committing to individual fates7–12. Here, single cell transcriptional profiling of zebrafish pigment cell development leads to us proposing a Cyclical Fate Restriction mechanism of NCC development that reconciles the DFR and PFR models. Our clustering of single NCC Nanostring transcriptional profiles identifies only broadly multipotent intermediate states between ENCCs and differentiated melanocytes and iridophores. Leukocyte tyrosine kinase (Ltk) marks the multipotent progenitor and iridophores, consistent with biphasic ltk expression13–15. Ltk inhibitor and constitutive activation studies support expression at an early multipotent stage, whilst lineage-tracing of ltk-expressing cells reveals their multipotency extends beyond pigment cell-types to neural fates. We conclude that pigment cell development does not involve a conventional PFR mechanism, but instead occurs directly and more dynamically from a broadly multipotent intermediate state.

The ability of small-cell wireless networks to self-organize is crucial for improving capacity and performance in modern communication networks. This paper considers one of the most basic questions: what is the expected distance to a cell’s nearest neighbour in a spatially distributed network? We analyse a model problem in the asymptotic limit of large total received signal and compare the accuracy of different heuristics. We also analytically consider the effects of fading. Our analysis shows that the most naive heuristic systematically underestimates the distance to the nearest node; this is substantially corrected in cases of interest by inclusion of the next-order asymptotic term. We illustrate our theoretical results explicitly or several combinations of signal and path loss parameters and show that our theory is well supported by numerical simulations.

The forecasting of sudden, irreversible shifts in natural systems is a challenge of great importance, whose realization could allow pre-emptive action to be taken to avoid or mitigate catastrophic transitions, or to help systems adapt to them. In recent years, there have been many advances in the development of such early warning signals. However, much of the current toolbox is based around the tracking of statistical trends and therefore does not aim to estimate the future time scale of transitions or resilience loss. Metric-based indicators are also difficult to implement when systems have inherent oscillations which can dominate the indicator statistics. To resolve these gaps in the toolbox, we use additional system properties to fit parsimonious models to dynamics in order to predict transitions. Here, we consider nearly-one-dimensional systems—higher dimensional systems whose dynamics can be accurately captured by one-dimensional discrete time maps. We show how the nearly one-dimensional dynamics can be used to produce model-based indicators for critical transitions which produce forecasts of the resilience and the time of transitions in the system. A particularly promising feature of this approach is that it allows us to construct early warning signals even for critical transitions of chaotic systems. We demonstrate this approach on two model systems: of phosphorous recycling in a shallow lake, and of an overcompensatory fish population.

Anodic passivation for copper exposed to aqueous NaCl (model seawater) is rate limited by diffusion of a poorly soluble Cu(I) chloro species. As a result, a protective layer of CuCl forms on copper metal (with approx. 1 μm thickness) that is then put under strain at more positive applied potentials with explosive events causing current spikes and particulate product expulsion. In this report, the mechanism for this explosive film rupture and particle expulsion process is shown to occur (i) in the absence of underlying anodic gas evolution, and (ii) linked to the presence/nature of gaseous solutes. The film rupture event is proposed to be fundamentally dependent on gas bubble nucleation (triggered by the release of interfacial stress) with surface tension effects by dissolved gases affecting the current spike pattern. Oxygen O2, hydrogen H2, and helium He suppress current spikes and behave differently to argon Ar, nitrogen N2, and carbon dioxide CO2, which considerably enhance current spikes. Vacuum-degassing the electrolyte solution results in behaviour very similar to that observed in the presence of helium. The overall corrosion rate for copper microelectrodes is compared and parameters linked to passivation and corrosion processes are discussed.

A growing literature considers the Sustainable Development Goals (SDGs) as a interlinked network, connected by co-benefits and trade-offs between pairs of SDGs. Such network descriptions naturally prompt important questions concerning the emergence and identification of system-level features. This paper develops mathematical techniques to address, quantitatively, the extent to which these interlinkage networks point to the likelihood of greater progress on some SDGs than on others, the sensitivity of the networks to the addition of new links (or the strengthening or weakening of existing ones), and the existence of implicit hierarchies within Agenda 2030. The methods we discuss are applicable to any directed network but we interpret them here in the context of three interlinkage matrices produced from expert analysis and literature reviews. We use these as three specific examples to discuss the quantitative results that reveal similarities and differences between these networks, as well as to comment on the mathematical techniques themselves. In broad terms, our findings confirm those from other sources, such as the Sustainable Development Solutions Network: for example, that globally SDGs 12–15 are most at risk. Perhaps of greater value is that analysis of the interlinkage networks is able to illuminate the underlying structural issues that lead to these systemic conclusions, such as the extent to which, at the whole-system level, the structure of SDG interlinkages favours some SDGs over others. The sensitivity analyses also suggest ways to quantify possible improvements to an SDG interlinkage network, since the sensitivity analyses are able to identify the modifications of the network that would best improve outcomes across the whole of Agenda 2030. This therefore indicates possibilities for informing policy-making, since the interlinkage networks themselves are implicitly descriptions of the overlaps, co-benefits and trade-offs that are anticipated to be likely to arise from a set of existing or proposed future policy actions.

Electrochemical studies have been conducted at copper microelectrodes (125, 50, and 25 μm in diameter) immersed in aqueous 0.5 M NaCl. Cyclic and linear sweep voltammetry were used to explore the corrosion of copper in chloride media. Cyclic voltammetry revealed the reversible Cu(I)/Cu(0) potential at approximately −0.11 V vs. SCE associated with the formation of a dense CuCl blocking layer (confirmed by in situ Raman and fluorescence measurements). Although continuous dissolution of Cu(I) occurs, only an increase in the driving potential into the region of the Cu(II)/Cu(I) potential at approximately +0.14 V vs. SCE started more rapid and stochastic dissolution/corrosion processes. The corrosion process is demonstrated to be linked to two distinct mechanisms based on (A) slow molecular dissolution and (B) fast colloidal dissolution. A polymer of intrinsic microporosity (PIM-EA-TB) is employed to suppress colloidal processes to reveal the underlying molecular processes.

Motivated by the propagation of thin bacterial films around planar obstacles, this paper considers the dynamics of travelling wave solutions to the Fisher–KPP equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_t = u(1-u) + u_{xx} + u_{yy}$$\end{document}ut=u(1-u)+uxx+uyy in a planar strip \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\infty< x < \infty $$\end{document}-∞0$$\end{document}α>0, at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=0$$\end{document}y=0. The presence of boundary conditions of this kind leads to the development of front solutions that propagate in x but contain transverse structure in y. Motivated by the observation that the speed of propagation in the Fisher–KPP equation is determined (for exponentially decaying initial conditions) by the behaviour at the leading edge, we analyse the linearised Fisher–KPP equation in order to estimate the speed of the stable travelling front, a function of the width L and the imposed boundary conditions. For wide strips the speed estimate based on the linearised equation agrees well with the results of numerical simulations. For narrow channels numerical simulations indicate that the stable front propagates more slowly, and for sufficiently small L or sufficiently large \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}α the front speed falls to zero and the front collapses. The reason for the collapse is the non-existence, far behind the front, of a stable positive equilibrium solution u(x, y). While existence of these equilibrium states can be demonstrated via phase plane arguments, the investigation of stability is similar to calculations of critical patch sizes carried out in similar ecological models.

Echo State Networks (ESNs) are a class of single-layer recurrent neural networks with randomly generated internal weights, and a single layer of tuneable outer weights, which are usually trained by regularised linear least squares regression. Remarkably, ESNs still enjoy the universal approximation property despite the training procedure being entirely linear. In this paper, we prove that an ESN trained on a sequence of observations from an ergodic dynamical system (with invariant measure μ ) using Tikhonov least squares regression against a set of targets, will approximate the target function in the L 2 ( μ ) norm. In the special case that the targets are future observations, the ESN is learning the next step map, which allows time series forecasting. We demonstrate the theory numerically by training an ESN using Tikhonov least squares on a sequence of scalar observations of the Lorenz system.

We analyse the model for vegetation growth in a semi-arid landscape proposed by von Hardenberg et al. (Phys. Rev. Lett. 87:198101, 2001), which consists of two parabolic partial differential equations that describe the evolution in space and time of the water content of the soil and the level of vegetation. This model is a generalisation of one proposed by Klausmeier but it contains additional terms that capture additional physical effects. By considering the limit in which the diffusion of water in the soil is much faster than the spread of vegetation, we reduce the system to an asymptotically simpler parabolic-elliptic system of equations that describes small amplitude instabilities of the uniform vegetated state. We carry out a thorough weakly nonlinear analysis to investigate bifurcations and pattern formation in the reduced model. We find that the pattern forming instabilities are subcritical except in a small region of parameter space. In the original model at large amplitude there are localised solutions, organised by homoclinic snaking curves. The resulting bifurcation structure is well known from other models for pattern forming systems. Taken together our results describe how the von Hardenberg model displays a sequence of (often hysteretic) transitions from a non-vegetated state, to localised patches of vegetation that exist with uniform low-level vegetation, to periodic patterns, to higher-level uniform vegetation as the precipitation parameter increases.

The physics of dry friction is often modelled by assuming that static and kinetic frictional forces can be represented by a pair of coefficients usually referred to as μs and μk, respectively. In this paper we re-examine this discontinuous dichotomy and relate it quantitatively to the more general, and smooth, framework of rate-and-state friction. This is important because it enables us to link the ideas behind the widely used static and dynamic coefficients to the more complex concepts that lie behind the rate-and-state framework. Further, we introduce a generic framework for rate-and-state friction that unifies different approaches found in the literature. We consider specific dynamical models for the motion of a rigid block sliding on an inclined surface. In the Coulomb model with constant dynamic friction coefficient, sliding at constant velocity is not possible. In the rate-and-state formalism steady sliding states exist, and analysing their existence and stability enables us to show that the static friction coefficient μs should be interpreted as the local maximum at very small slip rates of the steady state rate-and-state friction law. Next, we revisit the often-cited experiments of Rabinowicz (J. Appl. Phys., 22:1373–1379, 1951). Rabinowicz further developed the idea of static and kinetic friction by proposing that the friction coefficient maintains its higher and static value μs over a persistence length before dropping to the value μk. We show that there is a natural identification of the persistence length with the distance that the block slips as measured along the stable manifold of the saddle point equilibrium in the phase space of the rate-and-state dynamics. This enables us explicitly to define μs in terms of the rate-and-state variables and hence link Rabinowicz's ideas to rate-and-state friction laws. This stable manifold naturally separates two basins of attraction in the phase space: initial conditions in the first one lead to the block eventually stopping, while in the second basin of attraction the sliding motion continues indefinitely. We show that a second definition of μs is possible, compatible with the first one, as the weighted average of the rate-and-state friction coefficient over the time the block is in motion.

We study the phenomenon of cyclic dominance in the paradigmatic Rock--Paper--Scissors model, as occurring in both stochastic individual-based models of finite populations and in the deterministic replicator equations. The mean-field replicator equations are valid in the limit of large populations and, in the presence of mutation and unbalanced payoffs, they exhibit an attracting limit cycle. The period of this cycle depends on the rate of mutation; specifically, the period grows logarithmically as the mutation rate tends to zero. We find that this behaviour is not reproduced in stochastic simulations with a fixed finite population size. Instead, demographic noise present in the individual-based model dramatically slows down the progress of the limit cycle, with the typical period growing as the reciprocal of the mutation rate. Here we develop a theory that explains these scaling regimes and delineates them in terms of population size and mutation rate. We identify a further intermediate regime in which we construct a stochastic differential equation model describing the transition between stochastically-dominated and mean-field behaviour., 25 pages, 11 figures

Barab\'asi-Albert's `Scale Free' model is the starting point for much of the accepted theory of the evolution of real world communication networks. Careful comparison of the theory with a wide range of real world networks, however, indicates that the model is in some cases, only a rough approximation to the dynamical evolution of real networks. In particular, the exponent $\gamma$ of the power law distribution of degree is predicted by the model to be exactly 3, whereas in a number of real world networks it has values between 1.2 and 2.9. In addition, the degree distributions of real networks exhibit cut offs at high node degree, which indicates the existence of maximal node degrees for these networks. In this paper we propose a simple extension to the `Scale Free' model, which offers better agreement with the experimental data. This improvement is satisfying, but the model still does not explain \emph{why} the attachment probabilities should favor high degree nodes, or indeed how constraints arrive in non-physical networks. Using recent advances in the analysis of the entropy of graphs at the node level we propose a first principles derivation for the `Scale Free' and `constraints' model from thermodynamic principles, and demonstrate that both preferential attachment and constraints could arise as a natural consequence of the second law of thermodynamics.

Sliding frictional interfaces at a range of length scales are observed to generate travelling waves; these are considered relevant, for example, to both earthquake ground surface movements and the performance of mechanical brakes and dampers. We propose an explanation of the origins of these waves through the study of an idealized mechanical model: a thin elastic plate subject to uniform shear stress held in frictional contact with a rigid flat surface. We construct a nonlinear wave equation for the deformation of the plate, and couple it to a spinodal rate-and-state friction law which leads to a mathematically well-posed problem that is capable of capturing many effects not accessible in a Coulomb friction model. Our model sustains a rich variety of solutions, including periodic stick–slip wave trains, isolated slip and stick pulses, and detachment and attachment fronts. Analytical and numerical bifurcation analysis is used to show how these states are organized in a two-parameter state diagram. We discuss briefly the possible physical interpretation of each of these states, and remark also that our spinodal friction law, though more complicated than other classical rate-and-state laws, is required in order to capture the full richness of wave types.

Robust heteroclinic cycles (RHCs) arise naturally in collections of symmetric differential equations derived as dynamical models in many fields, including fluid mechanics, game theory and population dynamics. In this paper, we present a careful study of the complicated dynamics generated by small amplitude periodic perturbations of a stable robust heteroclinic cycle (RHC). We give a detailed derivation of the Poincare map for trajectories near the RHC, asymptotically correct in the limit of small amplitude perturbations. This reduces the nonautonomous system in R 3 to a 2D map. We identify three distinct dynamical regimes. The distinctions between these regimes depend subtly on different distinguished limits of the two small parameters in the problem. The first regime corresponds to the RHC being only weakly attracting: here we show that the system is equivalent to a damped nonlinear pendulum with a constant torque. In the second regime the periodically-perturbed RHC is more strongly attracting and the system dynamics corresponds to that of a (non-invertible or invertible) circle map. In the third regime, of yet stronger attraction, the dynamics of the return map is chaotic and no longer reducible to a one-dimensional map. This third regime has been noted previously; our analysis in this paper focusses on providing quantitative results in the first two regimes.

A frictional spring‐block system has been widely used historically as a model to display some of the features of two slabs in sliding frictional contact. Put elat, Dawes and Willis (2008) demonstrated that equations governing the sliding of two slabs co uld be approximated by spring‐block equations, and studied relaxation oscillations for two sla bs driven by uniform relative motion at their outer surfaces, employing this approximation. The present work revisits this problem. The equations of motion are first formulated exactly, with fu ll allowance for wave reflections. Since the sliding is restricted to be independent of positio n on the interface, this leads to a set of differential-difference equations in the time domain. Formal but systematic asymptotic expansions reduce the equations to differential equations. Truncation of the differential system at the lowest non-trivial order reproduces a classical spring-bl ock system, but with a slightly different “equivalent mass” than was obtained in the earlier work. Retention of the next term gives a new system, of higher order, that contains also some explicit effects of wave reflections. The smooth periodic orbits that result from the spring‐block system in the regime of instability of steady sliding are “decorated” by an oscillation whose period is related to the travel time of the waves across the slabs. The approximating differential system reproduces this effect with reasonable accuracy when the mean sliding velocity is not too far from the critical velocity for the steady state. The differential system also displays a period-doubling bifurcati on as the mean sliding velocity is increased, corresponding to similar behaviour of the exact differential-difference system.

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 Nonlinearity

We consider a novel asymptotic limit of model equations proposed to describe the formation of localized states in a vertically vibrated layer of granular material or viscoelastic fluid. In physical terms, the asymptotic limit is motivated by experimental observations that localized states (“oscillons”) arise when regions of weak excitation are nevertheless able to expel material rapidly enough to reach a balance with diffusion. Mathematically, the limit enables a novel weakly nonlinear analysis to be performed which allows the local depth of the granular layer to vary by $O(1)$ amounts even when the pattern amplitude is small. The weakly nonlinear analysis and numerical computations provide a robust possible explanation of past experimental results.

In the context of rate-and-state friction, we revisit the crossover between the creep and inertial regimes in the dynamics of a spring–block system as observed and described in the dry friction experiment of Heslot et al. (1994) and Baumberger et al. (1994) . We show that the transition between the quasi-static motion of a spring–block and its dynamic motion occurs at a lower sliding velocity than that which minimises the steady-state friction coefficient. We perform a weakly nonlinear stability analysis combined with numerical studies with the continuation package A uto . In particular, attention is focused on the change of nature the Hopf bifurcation from supercritical to subcritical, as observed by Heslot et al. Comparing the results obtained for different friction laws, we conclude that the weakly nonlinear analysis provides a possible criterion for distinguishing which friction laws may be physically relevant.

A subcritical pattern-forming (Turing) instability of a uniform state, in an infinite domain, produces two branches of spatially localized states that bifurcate from the pattern-forming instability along with a uniform spatially periodic pattern. In this paper we demonstrate that branches of localized states persist as strongly amplitude-modulated patterns in large, but finite, domains with periodic boundary conditions. Our analysis is carried out for a model Swift–Hohenberg equation with a cubic-quintic nonlinearity. If the domain size exceeds a critical value, modulated states appear in secondary bifurcations from the primary branch of spatially periodic solutions. Multiple-scales analysis indicates that these secondary bifurcations occur close to the primary instability and close to the saddle-node bifurcation on the spatially periodic solution branch. As the domain size increases, extra “turns” on the snaking curve arise through a repeating sequence of saddle-node bifurcations and mode interactions be...

We investigate the dynamics of pattern-forming systems in large domains near a codimension-two point corresponding to a ‘strong spatial resonance’ where competing instabilities with wavenumbers in the ratio 1 : 2 or 1 : 3 occur. We supplement the standard amplitude equations for such a mode interaction with Ginzburg–Landau-type modulational terms, appropriate to pattern formation in a large domain. In cases where the coefficients of these new diffusive terms differ substantially from each other, we show that spatially periodic solutions found near onset may be unstable to two long-wavelength modulational instabilities. Moreover, these instabilities generically occur near the codimension-two point only in the 1 : 2 and 1 : 3 cases, and not when weaker spatial resonances arise. The first instability is ‘amplitude-driven’ and is the analogue of the well-known Turing instability of reaction–diffusion systems. The second is a phase instability for which the subsequent nonlinear development is described, at leading order, by the Cahn–Hilliard equation. The normal forms for strong spatial resonances are also well known to permit uniformly travelling wave solutions. We also show that these travelling waves may be similarly unstable.

Steady states of localized activity appear naturally in uniformly driven, dissipative systems as a result of subcritical instabilities. In the usual setting of an infinite domain, branches of such localized states bifurcate at the subcritical “pattern-forming” instability and intertwine in a manner often referred to as “homoclinic snaking.” In this paper we consider an extension of this paradigm where, in addition to the pattern-forming instability (with nonzero wavenumber), a large-scale neutral mode exists, having zero growth rate at zero wavenumber. Such a situation naturally arises in the presence of a conservation law; we give examples of physical systems in which this arises, in particular, thermal convection in a horizontal fluid layer with a vertical magnetic field. We introduce a novel scaling that allows the derivation of a nonlocal Ginzburg–Landau equation to describe the formation of localized states. Our results show that the existence of the large-scale mode substantially enlarges the region...

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.

In the context of rate-and-state friction, we report an extensive analysis of stability of the quasi-static frictional sliding of two parallel interfaces dividing a linear elastic solid sheared at a constant rate. One possibility for the frictional sliding is that the interfaces slip at equal rates, a steady state described as symmetric. However a steady-state friction law that is non-monotonic allows the competing possibility of an asymmetric steady state in which the interfaces slide at different rates. A rate-and-state law that delivers such behaviour and agrees with the experimental results of Heslot et al. [1994. Creep, stick-slip, and dry-friction dynamics: experiments and a heuristic model. Phys. Rev. E 49, 4973–4988] is proposed. Analytical results combined with numerical investigations performed with the continuation package A uto and direct time integration are used to compile the complete picture of the many bifurcations that exist between the diverse steady and oscillatory sliding modes. In addition to the control parameters corresponding to the driving velocity and the stiffness of the medium, we find that the geometrical details of the steady-state friction law determine the occurrence and nature of bifurcations. Pitchfork bifurcations from the symmetric to asymmetric steady states coincide with the extrema of the friction law; Hopf bifurcations occur in the velocity weakening regime of the friction law. Torus and period-doubling bifurcations of periodic orbits also occur, and lead to complicated dynamics. We also present results of numerical computations that illustrate the complex and versatile dynamics of the two-interface system. We anticipate that the dynamics found in our model should be verifiable by experiments.

Thermal convection in a horizontal fluid layer heated uniformly from below usually produces an array of convection cells of roughly equal amplitudes. In the presence of a vertical magnetic field, convection may instead occur in vigorous isolated cells separated by regions of strong magnetic field. An approximate model for two-dimensional solutions of this kind is constructed, using the limits of small magnetic diffusivity, large magnetic field strength and large thermal forcing.The approximate model captures the essential physics of these localized states, enables the determination of unstable localized solutions and indicates the approximate region of parameter space where such solutions exist. Comparisons with fully nonlinear numerical simulations are made and reveal a power-law scaling describing the location of the saddle-node bifurcation in which the localized states disappear.

Robust heteroclinic cycles between equilibria lose stability either through local bifurcations of their equilibria or through global bifurcations. This paper considers a global loss of stability termed a `resonant' bifurcation. This bifurcation is usually associated with the birth or death of a nearby periodic orbit, and generically occurs in either a supercritical or subcritical manner. For a specific robust heteroclinic cycle between equilibria with complex eigenvalues we examine the codimension-two point that separates the supercritical and subcritical. We investigate the bifurcation structure and show the existence of further bifurcations of periodic orbits.

Heteroclinic networks are invariant sets containing more than one heteroclinic cycle. Such networks can appear robustly in equivariant vector fields. Previous authors have demonstrated that trajectories near heteroclinic networks can be attracted to one of a number of simultaneously 'stable' invariant subsets of the network. None of these invariant sets are asymptotically stable, but satisfy weaker definitions of stability. In this paper we discuss the behaviour of trajectories for one specific symmetric vector field. This vector field contains a robust heteroclinic network, and nearby trajectories display a variety of interesting dynamics. In particular, trajectories are observed to settle into a pattern of excursions around different parts of the network that we call 'cycling sub-cycles'. Cycling patterns displaying different numbers of loops around the individual component cycles can be stable for the same parameter values, as can combinations of regular and irregular cycling. Analytic results for the regular cycling behaviour agree well with numerical simulations. We show that there exist parameter values where some trajectories display irregular cycling behaviour, in the sense that the numbers of loops around individual sub-cycles form a bounded aperiodic infinite sequence.

Through multiple-scales and symmetry arguments we derive a model set of amplitude equations describing the interaction of two steady-state pattern-forming instabilities, in the case that the wavelengths of the instabilities are nearly in the ratio 1:2. In the case of exact 1:2 resonance the amplitude equations are ODEs; here they are PDEs. We discuss the stability of spatially-periodic solutions to long-wavelength disturbances. By including these modulational effects we are able to explore the relevance of the exact 1:2 results to spatially-extended physical systems for parameter values near to this codimension-two bifurcation point. These new instabilities can be described in terms of reduced `normal form' PDEs near various secondary codimension-two points. The robust heteroclinic cycle in the ODEs is destabilised by long-wavelength perturbations and a stable periodic orbit is generated that lies close to the cycle. An analytic expression giving the approximate period of this orbit is derived., 43 pages. Elsevier style files used

We study steady-state pattern-forming instabilities on 2. A uniform initial state that is invariant under the Euclidean group E(2) of translations, rotations and reflections of the plane loses linear stability to perturbations with a non-zero wavenumber kc. We identify branches of solutions that are periodic on a square lattice that inherits a reducible action of the symmetry group D4T2. Reducible group actions occur naturally when we consider solutions that are periodic on real-space lattices that are much more widely spaced than the wavelength of the pattern-forming instability. They thus apply directly to computations in large domains where periodic boundary conditions are applied. The normal form for the bifurcation is calculated, taking the presence of various `hidden' symmetries into account and making use of previous work by Crawford [8]. We compute the stability (relative to other branches of solutions that exist on this lattice) of the solution branches that we can guarantee by applying the equivariant branching lemma. These computations involve terms higher than third order in the normal form, and are affected by the hidden symmetries. The effects of hidden symmetries that we elucidate are relevant also to bifurcations from fully nonlinear patterns. In addition, other primary branches of solutions with submaximal symmetry are found always to exist; their existence cannot be deduced by applying the equivariant branching lemma. These branches are stable in open regions of the space of normal form coefficients. The relevance of these results is illustrated by numerical simulations of a simple pattern-forming PDE.

Techniques of equivariant bifurcation theory are used to study the Hopf bifurcation problem on a square lattice where the group Γ = D4×T2 acts Γ-simply on 8. This enables the analysis of the stability of solutions found in a previous analysis (Silber and Knobloch 1991 Nonlinearity 4 1063-106) of the Γ-simple representation on 4 to both solutions which are spatially periodic on a rhombic lattice, and to a countably infinite number of oscillatory `superlattice' solutions. The normal form for the bifurcation is computed, and conditions for the stability of all 17 -axial branches are given. Numerical investigations indicate that there exist open regions of coefficient space where the dynamics of the cubic order truncation of the normal form are chaotic. Chaotic dynamics have not previously been found in simpler Hopf bifurcation problems in normal form.

Rotating Boussinesq convection in a plane layer is governed by two dimensionless groups in addition to the Rayleigh number R: the Prandtl number σ and the Taylor number Ta. Scaled equations for fully nonlinear rotating convection in the limit of rapid rotation and small Prandtl number, where the onset of convection is oscillatory, are derived by considering distinguished limits where σnTa1/2 ∼ 1 but σ → 0 and Ta → ∞, for different n > 1. In the resulting asymptotic expansion in powers of Ta−1/2 and the amplitude of convection. Three distinct asymptotic regimes are identified, distinguished by the relative importance of the subdominant buoyancy and inertial terms. For the most interesting case, n = 4, the stability of different planforms near onset is investigated using a double expansion in powers of Ta−1/8 and the amplitude of convection ε. The lack of a buoyancy term at leading order demands that the perturbation expansion be continued through six orders to derive amplitude equations determining the dynamics. The case n = 1 is also analysed. The relevance of this theory to experimental results is briefly discussed.

We investigate the transition between oscillatory and steady convection at onset in low Prandtl number rotating convection in a plane layer. This transition is dominated by mode interactions which, at one point in parameter space, can be posed on a square lattice. This allows a rigorous reduction to a finite-dimensional bifurcation problem. We construct the normal form and compute the normal form coefficients for this codimension-2 bifurcation directly from the PDEs for Boussinesq rotating convection with stress-free upper and lower boundaries. The dynamics near the codimension-2 point are investigated fully; they explain behaviour found in numerical simulations of the PDEs at parameter values near the transition boundary. In particular, the normal form exhibits bursting dynamics created by a heteroclinic cycle containing points ‘at infinity’.

Three-dimensional pattern selection in a low Prandtl number Boussinesq fluid with stress-free boundaries, where the onset of convection is oscillatory, is explored. Restricting the problem to a square lattice, the normal form coefficients are calculated as functions of τ (the square root of the Taylor number) and the Prandtl number σ . There is a large region of the ( σ , τ ) plane where a heteroclinic cycle connecting four Travelling Roll states is stable. As σ is decreased the cycle undergoes a transverse loss of stability, creating quasiperiodic orbits which may themselves become chaotic. All these stable dynamics occur at onset. Although conjectured on the basis of general results from symmetric bifurcation theory (and well-known for steady convection as the Kuppers–Lortz instability [J. Fluid Mech. 35 (1969) 609]), cycling behaviour has not previously been demonstrated directly from the hydrodynamic equations in the oscillatory case. A second region of the ( σ , τ ) plane contains stable Travelling Roll solutions: we examine their stability to perturbations at varying angles and demonstrate the existence of small-angle instabilities of travelling rolls.

The first analysis of a properly three-dimensional mode interaction between steady and oscillatory forms of convection with different preferred wavenumbers is presented. By varying the fluid parameters for Boussinesq magnetoconvection we locate a point where the conduction state is unstable to both steady and oscillatory motion simultaneously. We then construct and analyse the normal form. The complex transition between steady and oscillatory convection near onset can be explained: this extends and completes the work of Clune and Knobloch (Pattern selection in three-dimensional magnetoconvection, Physica D 74 (1994) 151–176). Selecting the most marginal wavenumbers in the problem ensures that the analysis is relevant to the behaviour which would be observed in an unbounded plane layer. The symmetries of the resulting D 4 ⋉T 2 -equivariant bifurcation problem play a large role in determining the bifurcation structure and explain the appearance of interesting phenomena such as drifting solutions (A.M. Rucklidge, M. Silber, Bifurcations of periodic orbits with spatio-temporal symmetries, Nonlinearity 11 (1998) 1435–1455). We also find new phenomena in the normal form for a Hopf bifurcation with D 4 ⋉T 2 symmetry (M. Silber, E. Knobloch, Hopf bifurcation on a square lattice, Nonlinearity 4 (1991) 1063–1106). The introduction of weakly non-Boussinesq effects leads to qualitative changes in the dynamics near onset: different convection planforms are stabilised and chaotic heteroclinic cycling behaviour is observed.

The Hopf bifurcation with D 4 ⋉T 2 symmetry generically has open regions of the normal form coefficient space where all branches of periodic solutions bifurcate supercritically but none is stable [1] . In such regions we prove the existence of an attracting set near the origin. A new possibility for the attractor is a quasiperiodic solution branch related to Standing Cross Rolls (SCR). The new solution physically represents a planform we call Drifting Standing Cross Rolls . Unlike Standing Cross Rolls, this solution can be stable, as can a further triply-periodic solution. This explains the behaviour in the regions of coefficient space omitted by Silber and Knobloch [1] and completes their analysis.

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 D

We consider propagating, spatially localised waves in a class of equations that contain variational and non-variational terms. The dynamics of the waves is analysed through a collective coordinate approach. Motivated by the variational approximation, we show that there is a natural choice of projection onto collective variables for reducing the governing (nonlinear) partial differential equation (PDE) to coupled ordinary differential equations (ODEs). This projection produces ODEs whose solutions are exactly the stationary states of the effective Lagrangian that would be considered from applying the variational approximation method. We illustrate our approach by applying it to a modified Fisher equation for a travelling front, containing a non-constant-coefficient nonlinear term. We present numerical results that show that our proposed projection captures both the equilibria and the dynamics of the PDE much more closely than previously proposed projections., To appear in PRE (2013)

Recent work on the behavior of localized 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 nonvariational and nonconservative effects play a key role. Our work concentrates on aspects of this much more complicated problem. First we carry out the normal form analysis of the initial pattern-forming instability that leads to small-amplitude localized states. Next we examine the bifurcation structure of the large-amplitude localized states. Finally, we investigate the temporal stability of one-peak localized states. Throughout, we compare the localized states in the extended Swift-Hohenberg equation with the analogous solutions to the usual Swift-Hohenberg equation.

We present a reduced model for the transition to turbulence in shear flow that is simple enough to admit a thorough numerical investigation while allowing spatio-temporal dynamics that are substantially more complex than those allowed in previous modal truncations. Our model allows a comparison of the dynamics resulting from initial perturbations that are localised in the spanwise direction with those resulting from sinusoidal perturbations. For spanwise-localised initial conditions the subcritical transition to a `turbulent' state (i) takes place more abruptly, with a boundary between laminar and `turbulent' flow that is appears to be much less `structured' and (ii) results in a spatiotemporally chaotic regime within which the lifetimes of spatiotemporally complicated transients are longer, and are even more sensitive to initial conditions. The minimum initial energy $E_0$ required for a spanwise-localised initial perturbation to excite a chaotic transient has a power-law scaling with Reynolds number $E_0 \sim Re^p$ with $p \approx -4.3$. The exponent $p$ depends only weakly on the width of the localised perturbation and is lower than that commonly observed in previous low-dimensional models where typically $p \approx -2$. The distributions of lifetimes of chaotic transients at fixed Reynolds number are found to be consistent with exponential distributions., 22 pages. 11 figures. To appear in Proc. Roy. Soc. A

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.

The rate-and-state formulation of friction is well established as a phenomenological yet quantitative description of friction dynamics, in particular the onset of stick-slip instabilities arising from an oscillatory bifurcation. We first discuss the physical origins of two theories for the derivation of friction coefficients used in rate-and-state models, both derived from thermally activated rate processes. Secondly, we propose a general expression for the state evolution law in the form of a first order kinetics which describes the relaxation to a velocity dependent equilibrium interfacial state {symbol}ss (v) over a velocity dependent dynamic rejuvenation time-scale t{symbol} (v). We show that the unknown relation {symbol}ss (v), defined as the ratio of t{symbol} to a constant interfacial stationary healing time-scale t* *, can be estimated directly from the experimental measurements of the steady-state friction coefficient and the critical stiffness for the onset of stick-slip behaviour of a spring-block system. Using a specific experimental dataset, we finally illustrate that this method provides the experimental measurements of the apparent memory length La (v) = v t* * {symbol}ss (v) and the constant characteristic relaxation time t* * from which a constant intrinsic memory length L = V* t* * can be defined once a slip rate of reference V* is chosen. As a result the complete state evolution law can be experimentally characterised.

Coherent structures emerge from the dynamics of many kinds of dissipative, externally driven, nonlinear systems, and continue to provoke new questions that challenge our physical and mathematical understanding. In one specific sub-class of such problems, where a pattern-forming, or `Turing', instability occurs, rapid progress has been made recently in our understanding of the formation of localized states: patches of regular pattern surrounded by the unpatterned homogeneous background state. This short review article surveys the progress that has been made for localized states, proposes three areas of application for these ideas that would take the theory in new directions and ultimately be of substantial benefit to areas of applied science. Finally I offer speculations for future work, based on localized states, that may help researchers to understand coherent structures more generally., 14 pages, 6 figures. One contribution of 18 to a Triennial Issue `Visions of the future for the Royal Society's 350th anniversary year'

We consider localised states in a discrete bistable Allen-Cahn equation. This model equation combines bistability and local cell-to-cell coupling in the simplest possible way. The existence of stable localised states is made possible by pinning to the underlying lattice; they do not exist in the equivalent continuum equation. In particular we address the existence of 'isolas': closed curves of solutions in the bifurcation diagram. Isolas appear for some non-periodic boundary conditions in one spatial dimension but seem to appear generically in two dimensions. We point out how features of the bifurcation diagram in 1D help to explain some (unintuitive) features of the bifurcation diagram in 2D., Comment: 14 pages

International audience; The seismic cycle of earthquake recurrence is characterised by long periods of quasi-static evolution which precede sudden slip events accompanied by elastic wave radiation: the earthquake. This succession of processes over two well distinguished time scales recalls the behaviour of nonlinear relaxation oscillations. We explore this connection by studying, in the framework of rate-and-state friction, the sliding of two identical slabs of elastic solid driven in opposite directions with a constant relative velocity. Our first innovation is to establish that the motion of a spring-block system is an asymptotic mechanical analogue of the frictional sliding of a single interface from which elastic waves radiate. Due to wave reflection at the boundaries, the equivalent mass of the block $M=k(h/c_s)^2/12$ is not independent of the equivalent spring stiffness $k$, where $h$ denotes the slab thickness and $c_s$ is the shear wave speed. Considering a non-monotonic friction law, we show that the relaxation oscillation regime is reached when the characteristic time scale of frictionless oscillations is much greater than the characteristic time of frictional memory effects: $(M/k)^{1/2}\gg L/V_*$. We combine a composite approximation of the stick-slip cycle and numerical studies to show that the interfacial relaxation oscillations result from the subtle interplay of the non-monotonic properties of the friction law driving the long stress build-up of the quasi-static phase, and the inertial control of the fast slip phase originating from the wave propagation. We discuss the geophysical consequences for earthquake mechanics, and connections between the rate-and-state and Coulomb models of friction.

A viscous thread falling from a nozzle onto a surface exhibits the famous rope-coiling effect, in which the thread buckles to form loops. If the surface is replaced by a belt moving with speed $U$, the rotational symmetry of the buckling instability is broken and a wealth of interesting states are observed [See S. Chiu-Webster and J. R. Lister, J. Fluid Mech., {\bf 569}, 89 (2006)]. We experimentally studied this "fluid mechanical sewing machine" in a new, more precise apparatus. As $U$ is reduced, the steady catenary thread bifurcates into a meandering state in which the thread displacements are only transverse to the motion of the belt. We measured the amplitude and frequency $\omega$ of the meandering close to the bifurcation. For smaller $U$, single-frequency meandering bifurcates to a two-frequency "figure eight" state, which contains a significant $2\omega$ component and parallel as well as transverse displacements. This eventually reverts to single-frequency coiling at still smaller $U$. More complex, highly hysteretic states with additional frequencies are observed for larger nozzle heights. We propose to understand this zoology in terms of the generic amplitude equations appropriate for resonant interactions between two oscillatory modes with frequencies $\omega$ and $2\omega$. The form of the amplitude equations captures both the axisymmetry of the U=0 coiling state and the symmetry-breaking effects induced by the moving belt., Comment: 12 pages, 9 figures, revised, resubmitted to Physical Review E

Received 18 July 2006; published 6 November 2006Robust heteroclinic cycles occur naturally in many classes of nonlinear differential equations with invarianthyperplanes. In particular they occur frequently in models for ecological dynamics and fluid mechanicalinstabilities. We consider the effect of small-amplitude time-periodic forcing and describe how to reduce thedynamics to a two-dimensional map. In the limit where the heteroclinic cycle loses asymptotic stability,intervals of frequency locking appear. In the opposite limit, where the heteroclinic cycle becomes stronglystable, the dynamics remains chaotic and no frequency locking is observed.DOI: 10.1103/PhysRevE.74.055201 PACS number s : 05.45. a, 82.40.Bj