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Numerical continuation techniques are powerful tools that have been extensively used to identify particular solutions of nonlinear dynamical systems and enable trajectory design in chaotic astrodynamics problems such as the Circular Restricted Three-Body Problem. However, the applicability of equilibrium points and periodic orbits may be questionable in real-world applications where the uncertainties of the initial conditions of the spacecraft and dynamical parameters of the problem (e.g., mass ratio parameter) are taken into consideration. Due to uncertain parameters and initial conditions, the spacecraft might not follow the reference periodic orbit owing to growing uncertainties that cause the satellite to deviate from its nominal path. Hence, it is crucial to keep track of the probability of finding the spacecraft in a given region. Building on previous work, we extend numerical continuation to moments of the distribution (i.e., stochastic continuation) by directly continuing moments of the probability density function of the spacecraft state. Only assuming normality of the initial conditions, and leveraging moment-generating functions, Isserlis' theorem, and the algebra of truncated polynomials, we propagate the distribution of the spacecraft state at consecutive surface of section crossings while retaining a symbolic map of the final moments of the distribution that depend on the initial mean and covariance matrix only. The goal of the work is to offer a differential algebra-based general framework to continue 3D periodic orbits in the presence of uncertain dynamical systems. The proposed approach is compared against traditional Monte Carlo simulations to validate the uncertainty propagation approach and demonstrate the advantages of the proposed in terms of uncertainty propagation computational burden and access to higher-dimensional problems., Comment: Paper presented and published in the proceedings of the 2024 ISSFD Conference. (Darmstadt, Germany)

Data-driven modelling techniques provide a method for deriving models of dynamical systems directly from complicated data streams. However, tracking and forecasting such data streams poses a significant challenge to most methods, as they assume the underlying process and model does not change over time. In this paper, we apply one such data-driven method, the Koopman autoencoder (KAE), to high-dimensional oscillatory data to generate a low-dimensional latent space and model, where the system's dynamics appear linear. This allows one to accurately track and forecast systems where the underlying model may change over time. States and the model in the reduced order latent space can then be efficiently updated as new data becomes available, using data assimilation techniques such as the ensemble Kalman filter (EnKF), in a technique we call the KAE EnKF. We demonstrate that this approach is able to effectively track and forecast time-varying, nonlinear dynamical systems in synthetic examples. We then apply the KAE EnKF to a video of a physical pendulum, and achieve a significant improvement over current state-of-the-art methods. By generating effective latent space reconstructions, we find that we are able to construct accurate short-term forecasts and efficient adaptations to externally forced changes to the pendulum's frequency., Comment: 44 pages, 23 figures, 1 table

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., Comment: Review paper, 76 pages, 51 figures

Isolated patches of spatially oscillating pattern have been found to emerge near a pattern-forming instability in a wide variety of experiments and mathematical models. However, there is currently no mathematical theory to explain this emergence or characterise the structure of these patches. We provide a method for formally deriving radial amplitude equations to planar patterns via non-autonomous multiple-scale analysis and convolutional sums of products of Bessel functions. Our novel approach introduces nonautonomous differential operators, which allow for the systematic manipulation of Bessel functions, as well as previously unseen identities involving infinite sums of Bessel functions. Solutions of the amplitude equations describe fully localised patterns with non-trivial angular dependence, where localisation occurs in a purely radial direction. Amplitude equations are derived for multiple examples of patterns with dihedral symmetry, including fully localised hexagons and quasipatterns with twelve-fold rotational symmetry. In particular, we show how to apply the asymptotic method to the Swift--Hohenberg equation and general reaction-diffusion systems., Comment: 19 pages, 4 figures

Analytically tracking patterns emerging from a small amplitude Turing instability to large amplitude remains a challenge as no general theory exists. In this paper, we consider a three component reaction-diffusion system with one of its components singularly perturbed, this component is known as the fast variable. We develop an analytical theory describing the periodic patterns emerging from a Turing instability using geometric singular perturbation theory. We show analytically that after the initial Turing instability, spatially periodic patterns evolve into a small amplitude spike in the fast variable whose amplitude grows as one moves away from onset. This is followed by a secondary transition where the spike in the fast variable widens, its periodic pattern develops two sharp transitions between two flat states and the amplitudes of the other variables grow. The final type of transition we uncover analytically is where the flat states of the fast variable develop structure in the periodic pattern. The analysis is illustrated and motivated by a numerical investigation. We conclude with a preliminary numerical investigation where we uncover more complicated periodic patterns and snaking-like behaviour that are driven by the three transitions analysed in this paper. This paper provides a crucial step towards understanding how periodic patterns transition from a Turing instability to large amplitude., Comment: 38 pages, 13 figures

Data assimilation techniques, such as ensemble Kalman filtering, have been shown to be a highly effective and efficient way to combine noisy data with a mathematical model to track and forecast dynamical systems. However, when dealing with high-dimensional data, in many situations one does not have a model, so data assimilation techniques cannot be applied. In this paper, we use dynamic mode decomposition to generate a low-dimensional, linear model of a dynamical system directly from high-dimensional data, which is defined by temporal and spatial modes, that we can then use with data assimilation techniques such as the ensemble Kalman filter. We show how the dynamic mode decomposition can be combined with the ensemble Kalman filter (which we call the DMDEnKF) to iteratively update the current state and temporal modes as new data becomes available. We demonstrate that this approach is able to track time varying dynamical systems in synthetic examples, and experiment with the use of time-delay embeddings. We then apply the DMDEnKF to real world seasonal influenza-like illness data from the USA Centers for Disease Control and Prevention, and find that for short term forecasting, the DMDEnKF is comparable to the best mechanistic models in the ILINet competition.

Collective organisation of patterns into ring-like configurations has been well-studied when patterns are subject to either weak or semi-strong interactions. However, little is known numerically or analytically about their formation when the patterns are strongly interacting. We prove that approximate strongly interacting patterns can emerge in various ring-like dihedral configurations, bifurcating from quiescence near a Turing instability in generic two-component reaction-diffusion systems. The methods used are constructive and provide accurate initial conditions for numerical continuation methods to path-follow these ring-like patterns in parameter space. Our analysis is complemented by numerical investigations that illustrate our findings., Comment: 35 pages, 11 figures

Fully localised patterns involving cellular hexagons or squares have been found experimentally and numerically in various continuum models. However, there is currently no mathematical theory for the emergence of these localised cellular patterns from a quiescent state. A key issue is that standard techniques for one-dimensional patterns have proven insufficient for understanding localisation in higher dimensions. In this work, we present a comprehensive approach to this problem by using techniques developed in the study of axisymmetric patterns. Our analysis covers localised patterns equipped with a wide range of dihedral symmetries, avoiding a restriction to solutions on a predetermined lattice. The context in this paper is a theory for the emergence of such patterns near a Turing instability for a general class of planar reaction-diffusion equations. Posing the reaction-diffusion system in polar coordinates, we carry out a finite-mode Fourier decomposition in the angular variable to yield a large system of coupled radial ordinary differential equations. We then utilise various radial spatial dynamics methods, such as invariant manifolds, rescaling charts, and normal form analysis, leading to an algebraic matching condition for localised patterns to exist in the finite-mode reduction. This algebraic matching condition is nontrivial, which we solve via a combination of by-hand calculations and Gr\"obner bases from polynomial algebra to reveal the existence of a plethora of localised dihedral patterns. These results capture the essence of the emergent localised hexagonal patterns witnessed in experiments. Moreover, we combine computer-assisted analysis and a Newton-Kantorovich procedure to prove the existence of localised patches with 6m-fold symmetry for arbitrarily large Fourier decompositions. This includes the localised hexagon patches that have been elusive to analytical treatment., Comment: 60 pages, 15 figures

This paper investigates the existence of localised axisymmetric (radial) patterns on the surface of a ferrofluid in the presence of a uniform vertical magnetic field. We formally investigate all possible small-amplitude solutions which remain bounded close to the pattern's centre (the core region) and decay exponentially away from the pattern's centre (the far-field region). The results are presented for a finite-depth, infinite expanse of ferrofluid equipped with a linear magnetisation law. These patterns bifurcate at the Rosensweig instability, where the applied magnetic field strength reaches a critical threshold. Techniques for finding localised solutions to a non-autonomous PDE system are established; solutions are decomposed onto a basis which is independent of the radius, reducing the problem to an infinite set of nonlinear, non-autonomous ODEs. Using radial centre manifold theory, local manifolds of small-amplitude solutions are constructed in the core and far-field regions, respectively. Finally, using geometric blow-up coordinates, we match the core and far-field manifolds; any solution that lies on this intersection is a localised radial pattern. Three distinct classes of stationary radial solutions are found: spot A and spot B solutions, which are equipped with two different amplitude scaling laws and achieve their maximum amplitudes at the core, and ring solutions, which achieve their maximum amplitudes away from the core. These solutions correspond exactly to the classes of localised radial solutions found for the Swift-Hohenberg equation. Different values of the linear magnetisation and depth of the ferrofluid are investigated and parameter regions in which the various localised radial solutions emerge are identified. The approach taken in this paper outlines a route to rigorously establishing the existence of axisymmetric localised patterns in the future., Comment: 74 pages, 23 figures

Many complex systems occurring in the natural or social sciences or economics are frequently described on a microscopic level, e.g., by lattice- or agent-based models. To analyze the states of such systems and their bifurcation structure on the level of macroscopic observables, one has to rely on equation-free methods like stochastic continuation. Here, we investigate how to improve stochastic continuation techniques by adaptively choosing the parameters of the algorithm. This allows one to obtain bifurcation diagrams quite accurately, especially near bifurcation points. We introduce lifting techniques which generate microscopic states with a naturally grown structure, which can be crucial for a reliable evaluation of macroscopic quantities. We show how to calculate fixed points of fluctuating functions by employing suitable linear fits. This procedure offers a simple measure of the statistical error. We demonstrate these improvements by applying the approach in analyses of (i) the Ising model in two dimensions, (ii) an active Ising model, and (iii) a stochastic Swift-Hohenberg model. We conclude by discussing the abilities and remaining problems of the technique.

Stationary fronts connecting the trivial state and a cellular (distorted) hexagonal pattern in the Swift-Hohenberg equation with a quadratic-cubic nonlinearity are known to undergo a process of infinitely many folds as a parameter is varied, known as homoclinic snaking, where new hexagon cells are added to the core, leading to a region of infinitely-many, co-existing localized states. Outside the homoclinic snaking region, the hexagon fronts can invade the trivial state in a bursting fashion. In this paper, we use a far-field core decomposition to set up a numerical path-following routine to trace out the bifurcation diagrams of hexagon fronts for the two main orientations of cellular hexagon pattern with respect to the interface in the bistable region. We find for one orientation that the hexagon fronts can destabilize as the distorted hexagons are stretched in the transverse direction leading to defects occurring in the deposited cellular pattern. We then plot diagrams showing when the selected fronts for the two main orientations, aligned perpendicular to each other, are compatible leading to a hexagon wavenumber selection prediction for hexagon patches on the plane. Finally, we verify the compatibility criterion for hexagon patches in the Swift-Hohenberg equation with a quadratic-cubic nonlinearity and a large non-variational perturbation. The numerical algorithms presented in this paper can be adapted to general reaction-diffusion systems., Comment: 24 pages, 16 figures

In this paper, we carry out numerical bifurcation analysis of depinning of fronts near the homoclinic snaking region, involving a spatial stripe cellular pattern embedded in a quiescent state, in the two-dimensional Swift-Hohenberg equation with either a quadratic-cubic or cubic-quintic nonlinearity. We focus on depinning fronts involving stripes that are orientated either parallel, oblique and perpendicular to the front interface, and almost planar depinning fronts. We show that invading parallel depinning fronts select both a far-field wavenumber and a propagation wavespeed whereas retreating parallel depinning fronts come in families where the wavespeed is a function of the far-field wavenumber. Employing a far-field core decomposition, we propose a boundary value problem for the invading depinning fronts which we numerically solve and use path-following routines to trace out bifurcation diagrams. We then carry out a thorough numerical investigation of the parallel, oblique, perpendicular stripe, and almost planar invasion fronts. We find that almost planar invasion fronts in the cubic-quintic Swift-Hohenberg equation bifurcate off parallel invasion fronts and co-exist close to the homoclinic snaking region. Sufficiently far from the 1D homoclinic snaking region, no almost planar invasion fronts exist and we find that parallel invasion stripe fronts may regain transverse stability if they propagate above a critical speed. Finally, we show that depinning fronts shed light on the time simulations of fully localised patches of stripes on the plane. The numerical algorithms detailed have wider application to general modulated fronts and reaction-diffusion systems., Comment: 42 pages, 22 figures

We investigate the existence of stationary fronts in a coupled system of two sine-Gordon equations with a smooth, "hat-like" spatial inhomogeneity. The spatial inhomogeneity corresponds to a spatially dependent scaling of the sine-Gordon potential term. The uncoupled inhomogeneous sine-Gordon equation has stable stationary front solutions that persist in the coupled system. Carrying out a numerical investigation it is found that these inhomogeneous sine-Gordon fronts loose stability, provided the coupling between the two inhomogeneous sine-Gordon equations is strong enough, with new stable fronts bifurcating. In order to analytically study the bifurcating fronts, we first approximate the smooth spatial inhomogeneity by a piecewise constant function. With this approximation, we prove analytically the existence of a pitchfork bifurcation. To complete the argument, we prove that transverse fronts for a piecewise constant inhomogeneity persist for the smooth "hat-like" spatial inhomogeneity by introducing a fast-slow structure and using geometric singular perturbation theory.

We establish the existence of spatially localised one-dimensional free surfaces of a ferrofluid near onset of the Rosensweig instability, assuming a general (nonlinear) magnetisation law. It is shown that the ferrohydrostatic equations can be derived from a variational principle that allows one to formulate them as an (infinite-dimensional) spatial Hamiltonian system in which the unbounded free-surface direction plays the role of time. A centre-manifold reduction technique converts the problem for small solutions near onset to an equivalent Hamiltonian system with finitely many degrees of freedom. Normal-form theory yields the existence of homoclinic solutions to the reduced system, which correspond to spatially localised solutions of the ferrohydrostatic equations.

Steering a complex system towards a desired outcome is a challenging task. The lack of clarity on the system's exact architecture and the often scarce scientific data upon which to base the operationalisation of the dynamic rules that underpin the interactions between participant entities are two contributing factors. We describe an analytical approach that builds on Fuzzy Cognitive Mapping (FCM) to address the latter and represent the system as a complex network. We apply results from network controllability to address the former and determine minimal control configurations - subsets of factors, or system levers, which comprise points for strategic intervention in steering the system. We have implemented the combination of these techniques in an analytical tool that runs in the browser, and generates all minimal control configurations of a complex network. We demonstrate our approach by reporting on our experience of working alongside industrial, local-government, and NGO stakeholders in the Humber region, UK. Our results are applied to the decision-making process involved in the transition of the region to a bio-based economy., Comment: In Proceedings Cassting'16/SynCoP'16, arXiv:1608.00177

We study grain boundaries between striped phases in the prototypical Swift-Hohenberg equation. We propose an analytical and numerical far-field-core decomposition that allows us to study existence and bifurcations of grain boundaries analytically and numerically using continuation techniques. This decomposition overcomes problems with computing grain boundaries in a large doubly periodic box with phase conditions. Using the spatially conserved quantities of the time-independent Swift-Hohenberg equation, we show that symmetric grain boundaries must select the marginally zig-zag stable stripes. We find that as the angle between the stripes is decreased, the symmetric grain boundary undergoes a parity-breaking pitchfork bifurcation where dislocations at the grain boundary split into disclination pairs. A plethora of asymmetric grain boundaries (with different angles of the far-field stripes either side of the boundary) is found and investigated. The energy of the grain boundaries is then mapped out. We find that when the angle between the stripes is greater than a critical angle, the symmetric grain boundary is energetically preferred while when the angle is less than the critical angle, the grain boundaries where stripes on one side are parallel to the interface are energetically preferred. Finally, we propose a classification of grain boundaries that allows us to predict various non-standard asymmetric grain boundaries., Comment: 38 pages, 28 figures

A thin flat rectangular plate supported on its edges and subjected to in-plane loading exhibits stable post-buckling behaviour. However, the introduction of a nonlinear (softening) elastic foundation may cause the response to become unstable. Here the post-buckling of such a structure is investigated and several important phenomena are identified, including the transition of patterns from stripes to spots and back again. The interaction between these forms is of importance for understanding the possible post-buckling behaviours of this structural system. In addition, both periodic and some localized responses are found to exist as the dimensions of the plate are increased and this becomes relevant when the characteristic wavelengths of the buckle pattern are small compared to the size of the plate. Potential application of the model range from macroscopic industrial manufacturing of structural elements to the understanding of micro- and nano-scale deformations in materials., Comment: 22 pages, 11 figures, Accepted Proceedings of the Royal Society A

We report on localized patches of cellular hexagons observed on the surface of a magnetic fluid in a vertical magnetic field. These patches are spontaneously generated by jumping into the neighborhood of the unstable branch of the domain covering hexagons of the Rosensweig instability upon which the patches equilibrate and stabilise. They are found to co-exist in intervals of the applied magnetic field strength parameter around this branch. We formulate a general energy functional for the system and a corresponding Hamiltonian that provides a pattern selection principle allowing us to compute Maxwell points (where the energy of a single hexagon cell lies in the same Hamiltonian level set as the flat state) for general magnetic permeabilities. Using numerical continuation techniques we investigate the existence of localized hexagons in the Young-Laplace equation coupled to the Maxwell equations. We find cellular hexagons possess a Maxwell point providing an energetic explanation for the multitude of measured hexagon patches. Furthermore, it is found that planar hexagon fronts and hexagon patches undergo homoclinic snaking corroborating the experimentally detected intervals. Besides making a contribution to the specific area of ferrofluids, our work paves the ground for a deeper understanding of homoclinic snaking of 2D localized patches of cellular patterns in many physical systems., Comment: 22 pages, 16 figures. Accepted version Journal of Fluid Mechanics

We study localised activity patterns in neural field equations posed on the Euclidean plane; such models are commonly used to describe the coarse-grained activity of large ensembles of cortical neurons in a spatially continuous way. We employ matrix-free Newton-Krylov solvers and perform numerical continuation of localised patterns directly on the integral form of the equation. This opens up the possibility to study systems whose synaptic kernel does not lead to an equivalent PDE formulation. We present a numerical bifurcation study of localised states and show that the proposed models support patterns of activity with varying spatial extent through the mechanism of homoclinic snaking. The regular organisation of these patterns is due to spatial interactions at a specific scale associated with the separation of excitation peaks in the chosen connectivity function. The results presented form a basis for the general study of localised cortical activity with inputs and, more specifically, for investigating the localised spread of orientation selective activity that has been observed in the primary visual cortex with local visual input., Comment: 21 pages, 13 figures, submitted for peer review