arXiv subjects

David J. B. Lloyd

Publications and source records attributed to David J. B. Lloyd.

At least 19 recordsLinked to original sources

arXiv2026

The Game is the Game: Dynamic network analysis and shifting roles in criminal networks

Objectives: This paper incorporates time as a crucial variable to identify key players in criminal networks and explores how actors' positions change over time. It then assesses the accuracy of the results against the uncertainty around network data collected from criminal justice records. Methods: Network data are from a judicial document for a two-year investigation targeting a drug trafficking and distribution network. We use Katz centrality in its dynamic version to explore changes in relationships and relative importance of network actors. We then use a novel method of introducing new edges to the network using Bernoulli random trials to simulate missing data and assess the extent to which node rankings based on Katz centrality change or remain the same when introducing some level of uncertainty to our observed network. Results: We identify actors who consistently held a central role over the course of the two-year investigation and differentiate them from actors who provided key contributions to the group's activities, but only for a limited period. We show that compared to centrality measures commonly used in criminal network analysis, dynamic Katz centrality is helpful to differentiate individual contributions even among central nodes and explore individual trajectories over time, even when data are incomplete. Conclusions: This paper demonstrates the value of key player identification using temporal network data and offers an additional analytical tool to both organised crime scholars trying to capture the complex nature of criminal collaboration and law enforcement agencies aiming at identifying appropriate targets and disrupting criminal groups.

cs.SI↗

arXiv2026

Global bifurcation of localised 2D patterns emerging from spatial heterogeneity

We present a general approach to prove the existence, both locally and globally in amplitude, of fully localised multi-dimensional patterns in partial differential equations containing a compact spatial heterogeneity. While one-dimensional localised patterns induced by spatial heterogeneities have been well-studied, proving the existence of fully localised patterns emerging from a Turing instability in higher dimensions remains a key open problem in pattern formation. In order to demonstrate the approach, we consider the two-dimensional Swift--Hohenberg equation, whose linear bifurcation parameter is perturbed by a radially-symmetric potential function. In this case, the trivial state is unstable in a compact neighbourhood of the origin and linearly stable outside. We prove the existence of local bifurcation branches of fully localised patterns, characterise their stability and bifurcation structure, and then rigorously continue solutions to large amplitude via analytic global bifurcation theory. Notably, the primary bifurcating branch in the Swift--Hohenberg equation alternates between an axisymmetric spot and a non-axisymmetric `dipole' pattern, depending on the width of the spatial heterogeneity.

nlin.PS↗

arXiv2026

Heteroclinic connections between finite-amplitude periodic orbits emerging from a codimension two singularity

Heteroclinic connections between two distinct hyperbolic periodic orbits in conservative systems are important in a wide range of applications. On the other hand, it is theoretically challenging to find large amplitude connections from scratch and compute them numerically. In this paper, we use a codimension two singularity, in a family of periodic orbits, as an organizing center for the emergence of heteroclinic connections. A normal form is derived whose unfolding produces two distinct finite amplitude periodic orbits with an explicit heteroclinic connection. We also construct heteroclinic connections far from the singularity by numerical continuation, using two numerical strategies: shooting and the core-farfield decomposition. A key geometric tool in the numerics is cylindrical foliations for the stable and unstable manifolds and their intersection. We introduce a new property of heteroclinic connections - the action - and show it is an invariant along foliations, it has a jump at a surface of section, and it appears in a central way in the normal form theory. We find that the difference in asymptotic phase between minus and plus infinity is also a key property. The theory is applied to the Swift-Hohenberg equation, the nonlinear Schrodinger with fourth order dispersion, and coupled Boussinesq equations from water waves, all of which have an energy and action conservation law.

math.DS↗

arXiv2025

Influence network reconstruction from discrete time-series of count data modelled by multidimensional Hawkes processes

Identifying key influencers from time series data without a known prior network structure is a challenging problem in various applications, from crime analysis to social media. While much work has focused on event-based time series (timestamp) data, fewer methods address count data, where event counts are recorded in fixed intervals. We develop network inference methods for both batched and sequential count data. Here the strong network connection represents the key influences among the nodes. We introduce an ensemble-based algorithm, rooted in the expectation-maximization (EM) framework, and demonstrate its utility to identify node dynamics and connections through a discrete-time Cox or Hawkes process. For the linear multidimensional Hawkes model, we employ a minimization-majorization (MM) approach, allowing for parallelized inference of networks. For sequential inference, we use a second-order approximation of the Bayesian inference problem. Under certain assumptions, a rank-1 update for the covariance matrix reduces computational costs. We validate our methods on synthetic data and real-world datasets, including email communications within European academic communities. Our approach effectively reconstructs underlying networks, accounting for both excitation and diffusion influences. This work advances network reconstruction from count data in real-world scenarios.

math.DS↗

arXiv2025

Tracking and forecasting oscillatory data streams using Koopman autoencoders and Kalman filtering

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.

math.DS↗

arXiv2025

Localized Patterns

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.

nlin.PS↗

arXiv2024

Radial amplitude equations for fully localised planar patterns

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.

math.DS↗

arXiv2024

Stochastic Continuation of Trajectories in the Circular Restricted Three-Body Problem via Differential Algebra

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.

physics.space-ph↗

arXiv2024

Dihedral rings of patterns emerging from a Turing bifurcation

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.

math.DS↗

arXiv2023

Analysing transitions from a Turing instability to large periodic patterns in a reaction-diffusion system

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.

math.DS↗

arXiv2023

Combining Dynamic Mode Decomposition with Ensemble Kalman Filtering for Tracking and Forecasting

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.

math.DS↗

arXiv2023

Approximate localised dihedral patterns near a Turing instability

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 Gröbner 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.

math.DS↗

arXiv2021

Localised Radial Patterns on the Free Surface of a Ferrofluid

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.

math.DS↗

arXiv2021

Hexagon Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

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.

nlin.PS↗

arXiv2020

Adaptive stochastic continuation with a modified lifting procedure applied to complex systems

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.

nlin.AO↗

arXiv2019

Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

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.

nlin.PS↗

arXiv2019

Existence of stationary fronts in a system of two coupled wave equations with spatial inhomogeneity

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.

math.DS↗

arXiv2016

Pattern formation on the free surface of a ferrofluid: spatial dynamics and homoclinic bifurcation

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.

math.AP↗