arXiv subjects

Jason J. Bramburger

Publications and source records attributed to Jason J. Bramburger.

At least 19 recordsLinked to original sources

arXiv2026

The Spectral Skeleton of Chaos: Koopman Wave Packets on Poincaré Sections

A Poincaré section replaces a flow by a return map, but for a chaotic system this map is usually known only from sampled crossings. We show that coarse transport can be read directly from Koopman spectral data, without fitting the map. Measure-preserving EDMD retains the isometric structure; riggedDMD then approximates spectral measures and constructs finite regularized wave packets. Packet phase supplies a finite-resolution transport coordinate; low modulus marks a singular skeleton where the phase becomes ill-conditioned. We demonstrate the idea on the Rössler system, a 32-mode Kuramoto--Sivashinsky Galerkin system, and the forced Duffing oscillator. The packets yield coarse symbolic models on sections ranging from an almost one-dimensional curve to a visibly thick set. Their graphs organize observed low-period orbits and guide targeted searches for others. In Duffing Regime~II, a seven-region rule accounts for $91\%$--$94\%$ of filtered one-step transitions, while failures in the lowest retained modulus decile occur at $5.08$--$5.20$ times the overall rate. The packets are not Koopman eigenfunctions, nor are the regions exact Markov partitions. Together these computations show how spectral information beyond isolated eigenpairs can expose chaotic transport directly from trajectories.

nlin.CD↗

arXiv2026

Stability of localized solutions to lattice dynamical systems

Localized patterns are spatially confined structures that arise in lattice dynamical systems and play an important role in physics, biology, and materials science. While their existence and bifurcation structure are well-understood, the stability of these solutions remains largely unexplored, particularly in discrete and high-dimensional settings. In this work, we develop a general theoretical framework to analyze the spectral stability of localized steady states in one-dimensional and multi-dimensional rectangular lattices. Our approach leverages the properties of front and back solutions, combined with a discrete Evans function formulation, to characterize the spectrum of localized solutions. We prove that, for well-separated regions of localization, the Evans function asymptotically factorizes into contributions from the underlying fronts and backs, allowing explicit counting of unstable eigenvalues. This framework applies to solutions with single or multiple plateaus, including oscillatory and multi-pulse configurations. We illustrate the results on a real-valued cubic-quintic Ginzburg-Landau lattice, a prototypical Nagumo-type system, and provide numerical demonstrations of bifurcation structures and eigenvalue spectra.

nlin.PS↗

arXiv2026

Learning graphons from data: Random walks, transfer operators, and spectral clustering

Many signals evolve in time as a stochastic process, randomly switching between states over discretely sampled time points. Here we make an explicit link between the underlying stochastic process of a signal that can take on a bounded continuum of values and a random walk process on a graphon. Graphons are infinite-dimensional objects that represent the limit of convergent sequences of graphs whose size tends to infinity. We introduce transfer operators, such as the Koopman and Perron--Frobenius operators, associated with random walk processes on graphons and then illustrate how these operators can be estimated from signal data and how their eigenvalues and eigenfunctions can be used for detecting clusters, thereby extending conventional spectral clustering methods from graphs to graphons. Furthermore, we show that it is also possible to reconstruct transition probability densities and, if the random walk process is reversible, the graphon itself using only the signal. The resulting data-driven methods are applied to a variety of synthetic and real-world signals, including daily average temperatures and stock index values.

stat.ML↗

arXiv2026

Weighted Birkhoff Averages Accelerate Data-Driven Methods

Many data-driven algorithms in dynamical systems rely on ergodic averages that converge painfully slowly. One simple idea changes this: taper the ends. Weighted Birkhoff averages can converge much faster (sometimes superpolynomially, even exponentially) and can be incorporated seamlessly into existing methods. We demonstrate this with five weighted algorithms: weighted Dynamic Mode Decomposition (wtDMD), weighted Extended DMD (wtEDMD), weighted Sparse Identification of Nonlinear Dynamics (wtSINDy), weighted spectral measure estimation, and weighted diffusion forecasting. Across examples ranging from fluid flows to El Niño data, the message is clear: weighting costs nothing, is easy to implement, and often delivers markedly better results from the same data.

math.DS↗

arXiv2025

Reconstruction of frequency-localized functions from pointwise samples via least squares and deep learning

Recovering frequency-localized functions from pointwise data is a fundamental task in signal processing. We examine this problem from an approximation-theoretic perspective, focusing on least squares and deep learning-based methods. First, we establish a novel recovery theorem for least squares approximations using the Slepian basis from uniform random samples in low dimensions, explicitly tracking the dependence of the bandwidth on the sampling complexity. Building on these results, we then present a recovery guarantee for approximating bandlimited functions via deep learning from pointwise data. This result, framed as a practical existence theorem, provides conditions on the network architecture, training procedure, and data acquisition sufficient for accurate approximation. To complement our theoretical findings, we perform numerical comparisons between least squares and deep learning for approximating one- and two-dimensional functions. We conclude with a discussion of the theoretical limitations and the practical gaps between theory and implementation.

math.CA↗

arXiv2025

Bounding Escape Rates and Approximating Quasi-Stationary Distributions of Brownian Dynamics

Throughout physics Brownian dynamics are used to describe the behaviour of molecular systems. When the Brownian particle is confined to a bounded domain, a particularly important question arises around determining how long it takes the particle to encounter certain regions of the boundary from which it can escape. Termed the first passage time, it sets the natural timescale of the chemical, biological, and physical processes that are described by the stochastic differential equation. Probabilistic information about the first passage time can be studied using spectral properties of the deterministic generator of the stochastic process. In this work we introduce a framework for bounding the leading eigenvalue of the generator which determines the exponential rate of escape of the particle from the domain. The method employs sum-of-squares programming to produce nearly sharp numerical upper and lower bounds on the leading eigenvalue, while also giving good numerical approximations of the associated leading eigenfunction, the quasi-stationary distribution of the process. To demonstrate utility, the method is applied to prototypical low-dimensional problems from the literature.

math.OC↗

arXiv2025

Data-Driven Stabilisation of Unstable Periodic Orbits of the Three-Body Problem

Many different models of the physical world exhibit chaotic dynamics, from fluid flows and chemical reactions to celestial mechanics. The study of the three-body problem (3BP) and its many different families of unstable periodic orbits (UPOs) have provided fundamental insight into chaotic dynamics as far back as the 19th century. The 3BP, a conservative system, is inherently challenging to sample due to its volume-preservation property. In this paper we present an interpretable data-driven approach for the state-dependent control of UPOs in the 3BP, through leveraging the inherent sensitivity of chaos and the local manifold structure. We overcome the sampling challenge by utilising prior knowledge of UPOs and a novel augmentation strategy. This enables sample-efficient discovery of a verifiable and accurate local Poincaré map in as few as 55 data points. We suggest that the Poincaré map is best discovered at a surface of section where the norm of the monodromy matrix, i.e. the local sensitivity to small perturbations, is the smallest. To stabilise the UPOs, we apply small velocity impulses once each period, determined by solving a convex system of linear matrix inequalities based on the linearised map. We constrain the norm of the decision variables used to solve this system, resulting in locally optimal velocity impulses directed along the local stable manifold. Critically, this behaviour is achieved in a computationally efficient manner. We demonstrate this sample-efficient and low-energy method across several orbit families in the 3BP, with potential applications ranging from robotics and spacecraft control to fluid dynamics.

math.DS↗

arXiv2025

On the lattice property of the Koopman operator spectrum

The Koopman operator has become a celebrated tool in modern dynamical systems theory for analyzing and interpreting both models and datasets. The linearity of the Koopman operator means that important characteristics about it, and in turn its associated nonlinear system, are captured by its eigenpairs and more generally its spectrum. Many studies point out that the spectrum of the Koopman operator has a multiplicative lattice structure by which eigenvalues and eigenfunctions can be multiplied to produce new eigenpairs. However, these observations fail to resolve whether the new eigenfunction remains in the domain of the Koopman operator. In this work, we prove that the spectrum of the Koopman operator associated to discrete-time dynamical systems has a multiplicative lattice structure. We further demonstrate that the Koopman operator associated to discrete-time stochastic process does not necessarily have such a structure, demonstrating an important nuance that lies at the heart of Koopman operator theory.

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↗

arXiv2025

Real-Time Motion Detection Using Dynamic Mode Decomposition

Dynamic Mode Decomposition (DMD) is a numerical method that seeks to fit timeseries data to a linear dynamical system. In doing so, DMD decomposes dynamic data into spatially coherent modes that evolve in time according to exponential growth/decay or with a fixed frequency of oscillation. A prolific application of DMD has been to video, where one interprets the high-dimensional pixel space evolving through time as the video plays. In this work, we propose a simple and interpretable motion detection algorithm for streaming video data rooted in DMD. Our method leverages the fact that there exists a correspondence between the evolution of important video features, such as foreground motion, and the eigenvalues of the matrix which results from applying DMD to segments of video. We apply the method to a database of test videos which emulate security footage under varying realistic conditions. Effectiveness is analyzed using receiver operating characteristic curves, while we use cross-validation to optimize the threshold parameter that identifies movement.

cs.CV↗

arXiv2024

Transferability of Graph Neural Networks using Graphon and Sampling Theories

Graph neural networks (GNNs) have become powerful tools for processing graph-based information in various domains. A desirable property of GNNs is transferability, where a trained network can swap in information from a different graph without retraining and retain its accuracy. A recent method of capturing transferability of GNNs is through the use of graphons, which are symmetric, measurable functions representing the limit of large dense graphs. In this work, we contribute to the application of graphons to GNNs by presenting an explicit two-layer graphon neural network (WNN) architecture. We prove its ability to approximate bandlimited graphon signals within a specified error tolerance using a minimal number of network weights. We then leverage this result, to establish the transferability of an explicit two-layer GNN over all sufficiently large graphs in a convergent sequence. Our work addresses transferability between both deterministic weighted graphs and simple random graphs and overcomes issues related to the curse of dimensionality that arise in other GNN results. The proposed WNN and GNN architectures offer practical solutions for handling graph data of varying sizes while maintaining performance guarantees without extensive retraining.

cs.LG↗

arXiv2024

Complex localization mechanisms in networks of coupled oscillators: two case studies

Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial differential equations. While much of this understanding has been targeted at steady states, recent studies have noted complex dynamical localization phenomena in systems of coupled oscillators. These localized states can come in the form of symmetry-breaking chimera patterns that exhibit a coexistence of coherence and incoherence in symmetric networks of coupled oscillators and gap solitons emerging in the band gap of parametrically driven networks of oscillators. Here, we report detailed numerical continuations of localized time-periodic states in systems of coupled oscillators, while also documenting the numerous bifurcations they give way to. We find novel routes to localization involving bifurcations of heteroclinic cycles in networks of Janus oscillators and strange bifurcation diagrams resembling chaotic tangles in a parametrically driven array of coupled pendula. We highlight the important role of discrete symmetries and the symmetric branch points that emerge in symmetric models.

nlin.PS↗

arXiv2024

Persistence of steady-states for dynamical systems on large networks

The goal of this work is to identify steady-state solutions to dynamical systems defined on large, random families of networks. We do so by passing to a continuum limit where the adjacency matrix is replaced by a non-local operator with kernel called a graphon. This graphon equation is often more amenable to analysis and provides a single equation to study instead of the infinitely many variations of networks that lead to the limit. Our work establishes a rigorous connection between steady-states of the continuum and network systems. Precisely, we show that if the graphon equation has a steady-state solution whose linearization is invertible, there exists related steady-state solutions to the finite-dimensional networked dynamical system over all sufficiently large graphs converging to the graphon. The proof involves setting up a Newton--Kantorovich type iteration scheme which is shown to be a contraction on a suitable metric space. Interestingly, we show that the first iterate of our defined operator in general fails to be a contraction mapping, but the second iterate is proven to contract on the space. We extend our results to show that linear stability properties further carry over from the graphon system to the graph dynamical system. Our results are applied to twisted states in a Kuramoto model of coupled oscillators, steady-states in a model of neuronal network activity, and a Lotka--Volterra model of ecological interaction.

math.DS↗

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↗

arXiv2024

Data-driven Discovery of Invariant Measures

Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit model for the dynamics and allows one to target specific invariant measures, such as physical and ergodic measures. Moreover, it applies to both deterministic and stochastic dynamics in either continuous or discrete time. We provide convergence results and illustrate the performance of our method on data from the logistic map and a stochastic double-well system, for which invariant measures can be found by other means. We then use our method to approximate the physical measure of the chaotic attractor of the Rössler system, and we extract unstable periodic orbits embedded in this attractor by identifying discrete-time periodic points of a suitably defined Poincaré map. This final example is truly data-driven and shows that our method can significantly outperform previous approaches based on model identification.

math.DS↗

arXiv2023

Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization

We present a flexible data-driven method for dynamical system analysis that does not require explicit model discovery. The method is rooted in well-established techniques for approximating the Koopman operator from data and is implemented as a semidefinite program that can be solved numerically. Furthermore, the method is agnostic of whether data is generated through a deterministic or stochastic process, so its implementation requires no prior adjustments by the user to accommodate these different scenarios. Rigorous convergence results justify the applicability of the method, while also extending and uniting similar results from across the literature. Examples on discovering Lyapunov functions, performing ergodic optimization, and bounding extrema over attractors for both deterministic and stochastic dynamics exemplify these convergence results and demonstrate the performance of the method.

math.DS↗

arXiv2023

Synthesizing Control Laws from Data using Sum-of-Squares Optimization

The control Lyapunov function (CLF) approach to nonlinear control design is well established. Moreover, when the plant is control affine and polynomial, sum-of-squares (SOS) optimization can be used to find a polynomial controller as a solution to a semidefinite program. This letter considers the use of data-driven methods to design a polynomial controller by leveraging Koopman operator theory, CLFs, and SOS optimization. First, Extended Dynamic Mode Decomposition (EDMD) is used to approximate the Lie derivative of a given CLF candidate with polynomial lifting functions. Then, the polynomial Koopman model of the Lie derivative is used to synthesize a polynomial controller via SOS optimization. The result is a flexible data-driven method that skips the intermediary process of system identification and can be applied widely to control problems. The proposed approach is used to successfully synthesize a controller to stabilize an inverted pendulum on a cart.

math.OC↗

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↗