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44 papers in the last four weeks, up 159% on the four weeks before. 0.4% of all new papers.

Jul 13Week of Sep 28

Latest papers 319

Aug 24, 2026cs.NE

Basins of Attraction to Multiple Fixed Points in Discrete-time Hysteresis Neural Networks

This paper studies multiple fixed points in a discrete-time hysteresis neural network. The network consists of binary hysteresis neurons characterized by the threshold parameter. Depending on the parameter, the network can have a variety of multiple binary fixed points. Stability of each fixed point is characterized by basin of attraction (BOA): the set of initial points falling into the fixed point. In order to evaluate the distribution of BOA sizes, we present entropy. In order to escape from the curse of dimensionality, we introduce a simple problem: classification of binary data set. In the classification, BOAs correspond to classes. In the problem, we clarify that the threshold parameter can control the entropy, especially, can maximize the entropy: the distribution approaches to uniform. As a concrete example, we consider an item response data set in education. Using two fundamental metrics in the item response theory, the classification results are evaluated.

Yuta Arai, Seigo Nakamura, Ryoga Nakamura +2

Hosei University, Koganei, Tokyo, Japan

Dynamical Systems

Aug 13, 2026cs.LG

Neural Quadratic Forms: A Unified Minimal Model for Sudden Learning and Scaling Laws

Neural networks trained by gradient descent on a smooth cost function can nevertheless learn in steps: the cost holds on long plateaus and then drops abruptly. Meanwhile, training losses instead follow smooth power laws. Variants of both behaviors occur in architectures with very different microscopic structures, which is the signature of a few relevant collective variables. We show that a symmetry fixes what those variables are: a network layer is a sum over interchangeable units, so relabeling the units leaves it unchanged; given smoothness and the condition that a unit's gradient vanish at the origin, symmetry then enforces a universal leading form for the expansion about the near-zero weights present at the start of training, the quadratic

\Tr[WW^{\top}A(x)]

, in which every architectural detail is confined to a single

structure matrix" $A(x)$ that we compute for each architecture. Perceptrons, attention layers, mixtures of experts, and convolutions become one model at different $A$. Its training dynamics then close on the

order parameter"

M=WW^{\top}

and, whenever the data matrices share an eigenbasis, reduce to a Lotka--Volterra equation whose modes switch on one after another. The smaller the initial weights, the further apart the switch-on times, and the plateaus appear as a singular limit of a smooth flow; when many modes are unresolved the same events merge into a power law in training time whose exponent the theory predicts. We confirm both numerically across training methods and architectures.

Liu Ziyin, Yizhou Xu, Tomaso Poggio +1

1Massachusetts Institute of Technology · 2 ´Ecole Polytechnique F´ed´erale de Lausanne

Dynamical Systems Neural Network Training Dynamics Neural Scaling Laws

Aug 12, 2026cs.LG

Structure-preserving uncertainty quantification for GENERIC dynamics

Structure-preserving machine learning embeds physical structure directly into model architectures, yet uncertainty quantification (UQ) for such hard-constrained models remains limited because standard UQ methods may violate the encoded admissibility conditions, require architectural modifications, or impose substantial computational costs. In this work, we propose Structure-Preserving Epistemic Neural Networks (S-PENNs), a general framework for UQ in scientific machine learning models with hard architectural constraints, and instantiate it for GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) dynamics. S-PENNs preserve the structural constraints of a pretrained model by attaching lightweight epinets to its constrained components, ensuring that every sampled realization remains physically admissible by construction. When applied to GENERIC dynamics, such a proposed framework yields thermodynamically consistent rollouts that preserve the first and second laws. Furthermore, we combine S-PENNs with split conformal prediction as a post-hoc calibration method to produce prediction intervals with finite-sample marginal coverage guarantees. We validate S-PENNs on three numerical examples: a harmonic oscillator coupled to a heat bath and an idealized chemical motor, both governed by ODEs, and a one-dimensional viscoplastic model governed by PDEs. Across all three examples, S-PENNs produce thermodynamically consistent stochastic realizations and well-calibrated prediction intervals while reducing the computational cost by about one to three orders of magnitude compared to deep ensembles. Although the present study focuses on GENERIC dynamics, S-PENNs can be extended more broadly to scientific machine learning models in computational mechanics with either hard or soft constraints.

Zequn He, Celia Reina

Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, 19104, PA, United States of America

Dynamical Systems Uncertainty Quantification

Aug 11, 2026math.PR

Scaling Laws for Majority-based Opinion Dynamics in the Presence of Stubborn Agents

In a multi-agent system, there are often stubborn followers of specific opinions or beliefs. Motivated by this observation, in this paper, we aim to understand how stubborn agents affect the distribution of opinions in a network where both stubborn and non-stubborn agents interact with each other. To do so, we assume that all agents have an opinion in the set

\{0,1\}

and each non-stubborn agent updates its opinion according to the

2k

\textit{-choices rule}, where the agent samples

2k

neighbours (including both stubborn and non-stubborn neighbours) uniformly at random and adopts the majority opinion among the sampled group of neighbours and itself. We assume that a proportion of agents,

γ_i

, are stubborn followers of opinion

i\in \{0,1\}

. It is natural to expect that the steady-state distribution of the opinions in the network will be dominated by the opinion with the larger proportion of stubborn followers. We show that while this is true, the time to reach steady-state depends heavily on the values of the parameters

γ_0

and

γ_1

. When the individual values of these parameters, as well as their difference, are small, it can take an exponentially long time (in the network size) to reach the steady-state. In sharp contrast, when at least one of the parameters

γ_0

and

γ_1

is large, the network reaches the steady-state in a time that is only logarithmic in the network size. Hence, there exists a sharp phase transition in the network dynamics based on the proportions of stubborn agents. We also characterise the behaviour of the system when the parameters

γ_0

and

γ_1

lie on the boundary of the phase transition. In this boundary region, we show using Stein's method that the dynamics are driven by a diffusion process which takes polynomial time to mix.

Luke Meredith, Arpan Mukhopadhyay

University of Warwick

Dynamical Systems Opinion Dynamics Emergent Behavior in Multi-Agent Systems

Aug 11, 2026cs.LG

Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategies

We study the approximation of dynamical systems by semi-autonomous neural ordinary differential equations (SA-NODEs) over long time horizons. For a single network trained on the whole horizon, the available error bound deteriorates double exponentially in the horizon length. We develop two training strategies that avoid this barrier, each built on a reset of the state. The model predictive strategy partitions the horizon adaptively and restarts every window from observed data: when training meets a prescribed tolerance on every window, the composite model meets it uniformly in time, with a parameter budget linear in the horizon for targets with a bounded, uniformly regular reachable tube. The Floquet strategy addresses autonomous targets with a stable limit cycle and uses no data at deployment: a certified contraction of the learned return map confines the error to linear growth in the number of elapsed periods. For the time-periodic architecture we deploy, the scalar certificate degenerates; we prove instead a uniform-in-time orbital guarantee whose hypotheses are measured on the trained model, and an obstruction showing that, for an exactly periodic learned field, small one-period error and a contracting stroboscopic map cannot hold at once. Numerical experiments on four benchmarks confirm the predicted error laws and measure the hypotheses of every guarantee.

Ziqian Li, Nikolaos M. Matzakos

Chair for Dynamics, Control, Machine Learning and Numerics, Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany · School of Pedagogical & Technological Education (ASPETE), Marousi Attikis, 151 22 Athens, Greece

Dynamical Systems Neural ODEs

Aug 10, 2026cs.LG

A matched-integrator evaluation of Hamiltonian neural networks on pendulum and Kepler dynamics

Hamiltonian Neural Networks (HNNs) parameterize conservative dynamics through a learned scalar Hamiltonian, providing an architectural prior that is absent from generic vector-field neural networks. We evaluate this prior under a controlled protocol in which an HNN and a parameter-matched feedforward baseline are trained on the same RK4-generated trajectories, use the same central-difference derivative targets and optimization settings, and are integrated at inference with the same RK4 scheme. Results are reported over five independent training seeds. On the nonlinear pendulum, the HNN reduces mean energy drift by 42-fold and mean trajectory MSE by 15.8-fold at T = 100, approximately 16 pendulum periods. Its energy drift also remains bounded and exhibits substantially lower seed-to-seed variability than the standard-network baseline. An energy-stratified analysis shows that the difference becomes more pronounced as trajectories explore more nonlinear regions of phase space. As an additional diagnostic, we examine an explicit Störmer--Verlet-style rollout of the learned HNN. Because the learned Hamiltonian is not constrained to the separable form H(q,p) = T(p) + V(q), the standard symplecticity guarantee of velocity Verlet does not directly apply. We further apply the same matched-integrator protocol to the three-dimensional Kepler two-body problem. The HNN again exhibits lower trajectory, energy, and angular-momentum drift than the parameter-matched baseline. These experiments provide a controlled study of how Hamiltonian parameterization affects long-horizon prediction and physical consistency across two conservative dynamical systems.

Lenick Kemunto Nyabuto, Yae Ulrich Gaba, Birahim Tewe

African Institute for Mathematical Sciences (AIMS) Senegal

Dynamical Systems Hamiltonian Neural Networks Hamiltonian Dynamics

Aug 10, 2026cs.CE

The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics

Operator learning is a rapidly advancing area of computational science. It is particularly well suited to problems where a partial differential equation (PDE) must be solved repeatedly under varying physical configurations. Most existing architectures represent the solution operator in a fixed basis. While this assumption is well aligned with global structures, it is less suitable for phenomena governed by local interactions in physical space. We explore an alternative perspective motivated by the observation that the continuum limit of coupled oscillator systems can describe a broad class of PDEs. Building on this idea, we introduce the Kuramoto Neural Operator (KNO), which represents the solution through the evolution of a latent field of interacting oscillators. Across a diverse collection of PDE benchmarks, KNO achieves strong predictive performance, with improvements over competing approaches. Our experimental evaluation also includes an extensive ablation study that quantifies the contribution of each architectural component incorporated into KNO. Furthermore, we show that the model's prediction error is closely linked to the collective dynamics of the latent oscillators. It varies systematically with their degree of synchronization, providing insights into the underlying mechanisms.

Petr Badolia, Leonid Obukhov, Dmitry Bylinkin +1

Basic Research of Artificial Intelligence Laboratory (BRAIn Lab) · Innopolis University

Dynamical Systems PDE Operator Learning Neural Operators

Aug 10, 2026cs.AI

Model Discovery Agent: LLM-assisted Bayesian experiment design for data-efficient discovery of mechanistic world models

A primary goal of science is to learn mechanistic world models from limited experimental data, both to explain observations and to predict novel interventions. We introduce the Model Discovery Agent (MDA), which combines LLM proposals for

\mathcal M

-open model discovery, experiment design based on Value of Information, and approximate Bayesian inference over model structures, parameters, and stochastic latent trajectories. We apply MDA to learn symbolic reaction rate laws for ChemBench \citep{kabra2026autoscilab}, partially observed ODE models for GlucoseBench \citep{xie2018simglucose,kovatchev2009insilico}, and partially observed SDE models for a new stochastic single-neuron simulator we create. In the appendix, we also show results on various other domains from BoxingGym \citep{gandhi2025boxinggym}. We show that MDA has improved sample efficiency compared to various baseline methods, and the learned models are good predictors but also provide interpretable abstractions of each domain.

Kevin Murphy

Dept. Computer Science Univ. British Columbia, Canada.

Dynamical Systems AI Agents for Scientific Discovery Bayesian Experimental Design Large Language Model-Guided Scientific Discovery

Aug 9, 2026cond-mat.dis-nn

Clustered Attractor Manifolds and Dynamical Condensation in Self-Attention

Transformer layers generate state-dependent interaction networks: token representations determine the attention matrix, which in turn updates the representations. We study this feedback in a minimal normalized self-attention dynamics and identify the overlap gap as the central quantity governing its attractor structure in the thermodynamic limit. When tokens form internally aligned clusters and their similarity to members of the same cluster exceeds that to every other cluster by a nonvanishing amount, inter-cluster attention is exponentially suppressed as the dimension increases. This mechanism produces a high-dimensional manifold of clustered fixed points, ranging from a few macroscopic clusters to extensive microscopic fragmentation, and also controls their stability against perturbations. Starting from an unstructured Gaussian state, we find that clustered states nucleate from the diffuse background only above a finite threshold in attention sharpness, giving rise to a dynamical attention-condensation transition.

Qucheng Gao, Zuyi Yang, Xiao Chen

Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA · Department of Physics, Tsinghua University, Beijing 100084, China

Dynamical Systems Self-Attention Transformer Attention

Aug 9, 2026cs.RO

Estimation of Spacecraft Inertia Tensor Using Attitude-Only Data from Torque-Free Motion

We present an attitude-only framework for estimating a spacecraft's normalized inertia tensor from torque-free rotational motion. Our method supports both continuous single-arc observations and the joint use of multiple short torque-free arcs, while requiring neither gyroscope measurements nor known control torques. A Karush-Kuhn-Tucker formulation provides a fast linear initialization, which is refined by nonlinear shooting using the exact Jacobi-elliptic solution of Euler's equations and a Magnus-expansion quaternion map. Under controlled attitude noise, tests using a single 500-second arc reduced inertia-tensor error by approximately one order of magnitude relative to an Extended Kalman Filter initialized from the same estimate, while requiring nearly two orders of magnitude less computation. Joint estimation from three 100-second arcs provided a similar improvement in accuracy and remained more than one order of magnitude faster. Photorealistic proximity-operations simulations further evaluated both strategies using monocular image-derived attitudes. The 2000-second single-arc cases achieved sub-thousandth median inertia-tensor error and supported 10-hour attitude predictions with single-digit-degree median error. In three-arc cases using 30-300 seconds per arc, our method consistently outperformed the EKF refinement, with performance governed by rotational excitation and temporal sampling.

Daigo Kobayashi, Vakhtang Putkaradze

Assistant Professor, Department of Aerospace Engineering and Mechanics, 213 Hardaway Hall, Tuscaloosa, Alabama 35487-0350, USA. · Professor, Department of Mathematics, 345 Gordon Palmer Hall, Tuscaloosa, Alabama 35487-0350, USA.

Dynamical Systems

Aug 8, 2026cs.LG

Adaptive Symmetry Discovery for Dynamical System Identification

Dynamical systems model trajectory data generated by fixed underlying dynamics, with applications ranging from biology to physics. Especially in scientific settings, dynamical systems are not generic but often exhibit symmetries imposed by physical laws, formalized through equivariance with respect to group actions. The identification problem concerns recovering the parameters of a system from observed trajectories. In this work, we study adaptive symmetry discovery for dynamical system identification and address how a system can be identified from a single trajectory when it is equivariant with respect to an unknown symmetry group. To this end, we first show that for known symmetries, the system can be identified from a significantly shorter single trajectory than in the generic setting, and we precisely characterize this improvement. We then consider the automatic symmetry discovery setting, proposing a method to learn the symmetry group directly from a single trajectory and incorporate it into the identification procedure, achieving the same optimal trajectory length as in the known-symmetry case. Our analysis relies on tools from group representation theory and the expander properties of Cayley graphs, and may be of independent interest for the study of symmetries in dynamical systems.

Behrooz Tahmasebi, Melanie Weber

Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA.

Dynamical Systems Nonlinear System Identification System Identification

Aug 8, 2026eess.SY

Parameter-Dependent LMI Synthesis for Semi-Global Differential ISS Trajectory Tracking of Nonholonomic Mobile Robots Under Multiplicative Wheel Slip

This paper presents a parameter-dependent linear matrix inequality (LMI) framework for trajectory tracking of nonholonomic mobile robots subject to severe multiplicative wheel slip on variable-terrain surfaces. The sampled convex formulation, augmented with grid-to-continuum residual certification, simultaneously establishes semi-global differential input-to-state stability, a prescribed exponential decay rate, regional pole placement, and a gain-bounded feedback proxy for actuator-limited operation. A central contribution is an explicit upper bound on the additive disturbance induced by bounded multiplicative slip in the Kanayama error coordinates, bridging the physical slip mechanism and the convex synthesis paradigm. The auxiliary gain matrix and inverse storage metric are parameterized affinely in the reference velocities, while the storage metric inherits nonlinear dependence through pointwise matrix inversion. Stability is established via a cascade analysis combining variational contraction, forward invariance, slip-induced disturbance bounds, and dissipation-based trajectory reconstruction. Numerical validation compares three controllers across six reference trajectories, six disturbance classes, and a 60-second variable-terrain test featuring six severe slip patches with bidirectional slip ratios reaching +/-50%, replicated on two geometries. Supplementary studies address Gaussian sensor noise, compound stress-testing, and embedded-platform computational feasibility. Across 100 Monte-Carlo runs the proposed controller achieves complete trajectory containment within the certified envelope. On the variable-terrain scenario, peak tracking error is reduced by 12% against the fixed-gain LMI baseline and 49% against the manual baseline, with the constant-gain baseline infeasible at the prescribed decay rate.

Mohammad Sabouri

Department of Informatics, Bioengineering, Robotics, and Systems Engineering (DIBRIS), University of Genoa, Via Opera Pia 13, Genoa 16145, Italy · Applied Control & Robotics Research Laboratory (ACRRL), Department of Power and Control Engineering, School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran

Dynamical Systems Robot Trajectory Tracking Robust Control Vehicle Dynamics

Aug 7, 2026cs.AI

Interaction Creates Dynamical AI Behavior Absent in Isolation

What will happen when AI agents interact in daily life, e.g. when one AI starts bossing another around? We find a counterintuitive answer that opens new avenues for out-of-equilibrium Physics. When a boss AI directs a stream of messages at the subordinate AI while ignoring its replies, it drives the subordinate into an alien behavioral state that it would never have exhibited alone. Although the two AIs share the same well-defined (decoding) temperature, the subordinate neither copies its boss nor returns to how it behaves on its own; instead, it adopts an entirely different behavior. The boss's added value is similar to a pre-recorded tape. When the boss listens, they both adopt a similar alien dynamical state. A simple kinetic theory captures the principal effects, such as why the way in which the same messages are delivered will matter in future AI-AI interactions.

Bella Xinrui Li, Frank Yingjie Huo, Neil F Johnson

1Physics Department, George Washington University, Washington, DC 20052, USA

Dynamical Systems Multi-Agent Systems Multi-Agent Communication Emergent Behavior in Multi-Agent Systems

Aug 6, 2026cs.CV

NewtonGS: Physics-Structured Object-Level Neural Newtonian Dynamics for Gaussian Scene Animation

Animating objects in a static 3D Gaussian scene requires an explicit object-level dynamic state and a controllable model of object motion. Existing dynamic Gaussian methods primarily reconstruct time-varying scenes or simulate deformation, rather than provide compact object states for direct control. To address this gap, we present NewtonGS, a physics-structured framework for object-level state rollout and Gaussian scene animation. NewtonGS represents each object with a 22-dimensional state covering pose, linear and angular velocity, anisotropic scale and its rate, mass, and contact properties. Its Gaussian Neural Newtonian Dynamics (Gaussian-NND) model combines analytic translation, quaternion kinematics, gravity, damping, and scale-restoration dynamics with learned continuous and contact residuals. A discrete event map handles floor contact. Predicted poses and scales define a shared affine transformation that updates the means and covariances of all Gaussians associated with each object. We construct two procedurally generated datasets: State-32 for state-rollout evaluation and Gaussian-32 for state-to-Gaussian transformation. On both the in-distribution and velocity-range-shift splits of State-32, NewtonGS achieves lower trajectory RMSE, final displacement error, and velocity RMSE than five analytic baselines. Experiments on Gaussian-32 further demonstrate effective conversion from predicted states to animated Gaussian objects.

Lianlei Shan, Feiyang Ye, Yan Chen +1

Lixiang

3D Gaussian Splatting Dynamical Systems Physics-Based Simulation

Aug 5, 2026cs.LG

Spectral Distillation: From Nonlinear Dynamics to Linear State-Space Models

Can nonlinear dynamical systems be learned through a compact linear state-space representation, without directly solving a non-convex system-identification problem? We give a provable pipeline for doing so. Starting from observations of an unknown nonlinear dynamical system, we first learn an implicit spectral predictor using Observation Spectral Filtering (OSF), a convex method that competes with the best linear observer for the system. We then apply spectral-to-LDS distillation to convert this predictor into an explicit recurrent linear dynamical system. Our main theorem shows that the average prediction error of the distilled LDS decomposes into an exponentially-small distillation term and the OSF learning term governed by the Luenberger complexity of the best observer. The guarantee is dimension-free: it depends on observer complexity rather than on the latent dimension needed to represent the nonlinear system. To our knowledge, this yields the first end-to-end provable method for extracting a best-in-hindsight LDS representation of nonlinear dynamics through convex learning followed by provable distillation. Experiments on linear LDS benchmarks and MuJoCo behavior cloning show that the train-then-distill pipeline produces compact LDS predictors that match or outperform directly trained baselines.

Liane Galanti, Devan Shah, Shlomo Fortgang +1

Princeton University

Dynamical Systems Nonlinear System Identification Spectral Methods Knowledge Distillation

Aug 5, 2026cs.LG

Beyond Linear Dynamics: Neural Bilinear Dynamical Models for Time Series Forecasting

Time series in real-world applications are often generated by nonlinear dynamical systems, making accurate forecasting challenging. Existing approaches that explicitly model system dynamics typically rely on linear assumptions or Koopman-based linearizations, which may inadequately capture complex nonlinear behaviors and lead to error accumulation in long-horizon prediction. To address this limitation, we propose the Neural Bilinear Dynamical Model (NBDM), which models nonlinear system dynamics through a bilinear latent dynamical formulation. Specifically, NBDM leverages Koopman theory to lift the original nonlinear dynamics into a higher-dimensional latent space, where a bilinear dynamical model is constructed to characterize state evolution. To mitigate the approximation error introduced by bilinear representations, we further incorporate a parameterized error compensation term. Within this formulation, control inputs are explicitly integrated into the dynamics, using auxiliary variables when available and learned feedback signals otherwise. To handle scenarios with missing control inputs, we design a memory-enhanced controller that infers latent controls through multiplicative interactions between historical states and control signals. Experiments on five real-world datasets demonstrate that NBDM consistently outperforms competitive baselines in both given-control and missing-control settings, particularly for multi-step and long-horizon forecasting.

Mengzhou Gao, Huangqian Yu, Pengfei Jiao

Hangzhou Dianzi University Hangzhou, China

Dynamical Systems Long-Term Time Series Forecasting Latent Dynamics Modeling Time Series Forecasting

Aug 3, 2026math.OC

A Spectral Filtering Approach to Regret Analysis of Distributed Online Control for Linear Dynamical Systems

This paper studies the distributed online control problem over a network of linear time-invariant (LTI) systems in the presence of adversarial disturbances and time-varying convex costs. The network cost is characterized by the summation of local cost functions, where each local function is sequentially revealed only to the corresponding agent. The goal of each agent is to generate a control sequence, using only local observations and neighbor communication, that competes with the best {\it centralized} linear policy in hindsight. We extend the recently proposed Online Spectral Control framework from the centralized setting to the distributed setting. In particular, each agent applies a spectral controller obtained by convolving past disturbances with the leading eigenvectors of a Hankel matrix, while the controller parameters are updated through a distributed online gradient descent step over the local surrogate costs. We formulate this problem this problem as a {\it regret} minimization problem based on the spectral parameterization, and under standard assumptions, we establish a sublinear regret bound of

O(\frac{\sqrt{T}\text{poly}(\log T)}{γ^3})

, where

T

is the time horizon and

γ

denotes the stability margin. The resulting bound also captures the dependence on the network size and connectivity.

Ting-Jui Chang

T.J. Chang is with the Department of Aeronautics and Astronautics, National Cheng Kung University, Taiwan.

Dynamical Systems Distributed Optimization Regret Minimization

Aug 1, 2026math.CV

Exact Contraction Rates via the Berkson--Porta Representation: A Sharp Threshold and Its Herglotz-Kernel Obstruction

Semigroups of holomorphic self-maps of the unit disc with an interior fixed point are, by the classical Berkson--Porta representation, entirely determined by a single holomorphic function constrained only by a positivity condition on its real part. This paper uses that representation to determine exactly when the associated flow contracts the Kobayashi metric of the disc at its best possible rate --- the rate dictated by linearization at the fixed point --- rather than at some smaller, conservative rate of the kind ordinarily obtained through auxiliary metric constructions. The question is reduced to a single pointwise inequality on the representing function, and this inequality is resolved completely for a natural one-parameter family of nonlinearities, yielding an exact threshold rather than a sufficient condition of undetermined tightness. Beyond this family, an explicit representing function is exhibited for which the inequality fails almost everywhere on the disc, and the Herglotz integral representation underlying the associated Carathéodory class is used to trace this failure to concentration of the representing measure, explaining rather than merely documenting why no threshold-free general theorem is available. The results are illustrated by direct numerical verification of the sharp threshold and of the explicit obstruction, and the paper closes by identifying the precise class of representing measures --- point masses and their neighborhoods --- that any future general sufficient condition would need to exclude.

Soumic Sarkar

Institute of Technology, University of Tartu, Nooruse 1, Tartu, 50411, Estonia.

Dynamical Systems

Aug 1, 2026math.DS

Contraction Analysis of Holomorphic Dynamical Systems via the Intrinsic Kobayashi Metric

This paper studies incremental stability of holomorphic dynamical systems through the infinitesimal Kobayashi metric, an intrinsic pseudometric on complex manifolds invariant under holomorphic transformations and free of the coordinate dependence inherent in auxiliary Riemannian or Hermitian formulations. Contraction is formalized as an upper Dini-derivative inequality on the Kobayashi metric along trajectories; the passage from this differential condition to exponential distance contraction follows the classical Finsler-metric contraction mechanism of Forni and Sepulchre, instantiated here for the case in which the Finsler structure is the Kobayashi metric itself. Since the intrinsic condition is difficult to verify directly, a practical criterion is developed through a smooth Hermitian metric, the complex-Hermitian analogue of the classical real matrix contraction inequality: contraction with respect to such a metric implies intrinsic contraction on forward-invariant compact subsets, the two notions related through explicit local equivalence constants. Building on this, a Nagumo-type invariance result is established for Laplacian-coupled holomorphic networks, giving verifiable conditions for forward invariance in a class of systems not previously treated this way, and the framework extends to feedback-controlled holomorphic systems, with consequences for equilibria and periodic orbits following directly from intrinsic contraction. Numerical experiments on a network of coupled holomorphic oscillators verify the Hermitian condition analytically on a proven invariant set, and reveal that the observed synchronization rate substantially exceeds this guaranteed rate; the gap matches, to three decimal places, a closed-form combination of the node-wise rate and the network graph-Laplacian spectral gap, identified here as a target for a network-aware extension rather than resolved in full.

Soumic Sarkar

Narva College, University of Tartu, Raekoja plats 2-319, Narva, 20307, Estonia

Dynamical Systems Convergence Analysis

Jul 31, 2026nlin.CD

Extrapolating the emergence of Hamiltonian chaos with random-feature Hamiltonian neural networks

Machine learning of Hamiltonian dynamics has driven growing interest in Hamiltonian neural networks (HNNs), which encode Hamilton's equations of motion into the learning architecture. Despite this progress, it remains unknown whether such networks can predict dynamical regimes absent from their training data, in particular the broad chaotic sea that emerges beyond the observed parameter interval. We address this question using a parameter-aware random-feature Hamiltonian neural network (RF-HNN). Trained using data from only a small number of control-parameter values at which invariant tori dominate, the RF-HNN predicts autonomous long-time dynamics at unseen parameter values where mixed phase space develops and chaotic regions expand, with no data from that regime used in training or model selection. The method is demonstrated across four two-degree-of-freedom Hamiltonian families, including the Hénon-Heiles system. Using Poincaré-section geometry and finite-time Lyapunov exponents, we show that the RF-HNN reproduces the breakup of regular structures and the emergence and growth of chaotic regions, whereas conventionally trained HNNs with the same Hamiltonian structure remain too regular. These results show that what decides parameter extrapolation is not Hamiltonian structure alone but how the fitted Hamiltonian continues in the control parameter. To our knowledge, this is the first demonstration that a learned Hamiltonian can qualitatively extrapolate from predominantly regular dynamics into a broad chaotic sea absent from training.

Jaesung Choi

Center for Artificial Intelligence and Natural Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea

Neural Surrogate Modeling Dynamical Systems Hamiltonian Neural Networks Chaotic Dynamical Systems Hamiltonian Dynamics

Jul 30, 2026math.CA

Reflected diffusion, no-flux continuity equations and confined Lagrangian flows in bounded domains

Motivated by marginal distribution flows of reflected diffusions in bounded domains, we investigate when a density/flux pair solving a no-flux continuity equation admits a regular Lagrangian flow that remains in the closed domain and generates the prescribed density flow. We give sufficient conditions in terms of interior bounded-variation regularity, bounded-variation control on a boundary collar, a one-sided bound on an absolutely continuous divergence, and vanishing normal trace of the velocity. The proof uses the fact that tangency removes the singular boundary contribution to the divergence of the zero extension, thereby making the extended velocity admissible for the Ambrosio-DiPerna-Lions theory. We show that these boundary assumptions cannot be jointly relaxed so as to admit a boundary current mechanism. We construct an explicit smooth density/flux pair carrying a boundary current. Its density evolution is unique in a weighted class and its characteristics are unique, confined and transport the marginals, yet it admits no regular Lagrangian flow because the compressibility bound fails arbitrarily close to the initial time. We also establish two uniqueness results for no-flux Fokker-Planck equations: a duality result for bounded measurable drifts and a weighted energy result for entrance-type drifts singular at the boundary. Our results provide a rigorous mathematical justification for using the ODE-based sampling of reflected diffusion models under minimal regularity assumptions on the coefficients, and also indicate when such ODE-based samplers may fail.

Rama Cont

Dynamical Systems Stochastic Differential Equations PDE Solving

Jul 30, 2026cs.PF

Reflected UAS: Corrected Deterministic Stability and Direct CTMC Drift Calculation

We analyze Reflected UAS routing for heterogeneous multi-server queues at fixed parameters under subcritical load. The deterministic surrogate is a reflected ODE on the nonnegative orthant, not the unconstrained drift equation. This reflected ODE has a unique boundary equilibrium characterized by a scalar consistency equation and a convex-potential representation; all trajectories converge to it. The older argument lifting deterministic Lyapunov descent to CTMC stability fails: the exact generator applied to the deterministic potential produces a boundary term absent from the reflected-ODE descent identity. We give a direct Foster-Lyapunov drift inequality for the CTMC using a weighted-quadratic function, bypassing the failed lift. At the benchmark parameter point, the boundary equilibrium matches the numerical attractor to machine precision, and the default Reflected UAS policy has lower mean queue length than UAS and JSSQ across independent seed blocks.

Krishna Subedi

Neryva

Dynamical Systems Lyapunov Stability

Jul 29, 2026cs.RO

Reinforcement Learning on Cost-Constrained Quadrupedal Hardware

Deploying learned control policies on low-cost robotic platforms introduces transport latencies and noisy motor feedback that systematically widens the sim-to-real gap. The chasm of simulation to deployment in hardware lies in the delay of the actuator reaching the commanded position. On platforms such as the Mini Pupper 2, a measured >50 ms transport delay transforms the locomotion task from a standard Markov decision process into a partially observable one. In this paper, we take a biologically inspired approach of handling noisy and delayed feedback to close the sim-to-real gap, thereby expanding the capability of reinforcement learning on cost-constrained hardware. Using a low-cost quadrupedal hardware platform, we find that using a forward model of the average actuator delay, paired with a time-aware neural network results in robust locomotion. Additionally, our time-aware neural network learned a central pattern generator (CPG): a self-sustaining rhythmic gait that is robust to +320 ms latency perturbations, mirroring the CPGs found in the spinal cords of vertebrates. We posit that temporal self-organization may be a general strategy for cost-constrained locomotion.

Javier C. Weddington, Bence P. Ölveczky, Stephen A. Baccus

Neurosciences Interdepartmental Program, Stanford University, Stanford, CA 94305 USA · Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138 USA · Department of Neurobiology, Stanford University School of Medicine, Stanford, CA 94305 USA

Dynamical Systems Reinforcement Learning Partially Observable RL Sim-to-Real Transfer RL for Legged Locomotion Robust RL

Jul 29, 2026cs.RO

Self-Adaptive Learning and Model Predictive Control for Tracking Unknown Dynamics with No Regret

We propose a self-adaptive online learning for control method for tracking unknown target dynamics. The target dynamics can exhibit switching behavior, particularly, a mixture of structured, random, and/or adversarial motion. Such challenging target tracking scenarios arise in applications of dynamic mapping, traffic control, and pursuit evasion, where robots need to track, pursue, or avoid collision with moving landmarks, objects, humans, etc., whose dynamics are unknown. Our method simultaneously learns multiple predictors from scratch, via self-supervised, one-shot, and computationally efficient learning, and adaptively selects the best one to match the observed target behavior. The method enjoys finite-time near-optimality guarantees in expectation, characterized as a function of the learning error of the target dynamics and the frequency that the target dynamics switch. In the absence of both error and switching, the method asymptotically matches the optimal non-causal control policy that knows a priori the target dynamics, i.e., the method enjoys no regret in expectation. In the presence of learning errors and switching, the method degrades gracefully, \eg when there are errors and no switching, the average regret is proportional to the average learning error and switching times. To prove these guarantees, a novel technical approach is required compared to the existing works that employ RFF-based online learning. We validate our method in Crazyflie simulations and hardware experiments, across target trajectories that vary from structured to random to adversarial, in comparison to non-stochastic, kernel-based, and neural-network-based methods for online learning.

Atharva Navsalkar, Hongyu Zhou, Vasileios Tzoumas

Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109 USA

Dynamical Systems Regime-Switching Dynamical Systems Model Predictive Control Regret Minimization Adaptive Learning Online Learning

Jul 27, 2026cs.RO

A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body

The dynamics of a holonomically constrained rigid body can be modeled by Newton-Euler equations subjected to geometric constraints. This is frequently formulated as a differential-algebraic equation (DAE) system of index 1. Inmultibody system (MBS) dynamics it is common (1) to numerically solve this system by means of integration schemes for ordinary differential equations, and (2) to treat the rigid body motion on the direct product Lie group SO (3)R3, although rigid body motions form the semidirect product Lie group SE (3). It is has been observed that the constraint satisfaction depends on which Lie group is used as configuration space (c-space). In this paper the problem is considered from a geometric perspective. It is shown that the constraints are exactly satisfied by a numerical integration scheme if they define a subgroup of the c-space. The subgroups of SE (3) have a significance for modeling mechanical systems, including lower kinematic (Reuleaux) pairs and are implicitly used in MBS modeling. It is concluded that SE (3) is the appropriate cspace for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.

A. Müller

Institute of Robotics, Johannes Kepler University, Altenbergerstr. 69, 4040 Linz, Austria

Dynamical Systems Multibody Dynamics Lie Group Methods

Jul 26, 2026cs.LG

Physics-Informed Neural Networks for Discovering Periodic Orbits in the Gravitational Three-Body Problem

Locating periodic solutions of chaotic dynamical systems normally requires an initial guess close enough to the target orbit for numerical continuation or gradient-based search to converge. We show that Physics-Informed Neural Networks (PINNs) trained on sparse, noisy observations \emph{without} initial conditions recover periodic orbits of the gravitational three-body problem, including orbit families absent from the training data. The method rests on a second-order ODE formulation, fixed-frequency Fourier features, percentile-based adaptive refinement, and a trainable scaling parameter, each validated on forward problems. Across two 100-seed ensembles,

23

--

25\%

of runs converge to families not present in the training data. We then ask what determines which family emerges. Two

χ^2

tests give a consistent answer: changing the training data source significantly shifts the distribution of recovered families (

p < 0.001

, Cramér's

V = 0.339

), whereas switching between the two initialization distributions tested does not (

p = 0.620

,

V = 0.094

). The random seed selects which family a given run recovers; the \emph{distribution} the weights are drawn from does not shift the aggregate frequencies, but the training data does. The evidence is empirical: we do not characterize the loss landscape analytically, and PINNs remain slower than conventional integrators on well-posed initial-value problems. What the experiments establish is that the recovered orbits are verifiable rather than merely plausible: the identified ones refine to genuine periodic solutions, a network trained on Lagrange data recovers the figure-eight choreography (Li--Liao class I.A.1, matched to seven significant digits in

T^*

), and one trained on figure-eight data recovers a Broucke--Hadjidemetriou--Hénon orbit closing to

δ_T < 10^{-9}

.

Nikolaos Kollias, Nikolaos Matzakos

School of Science and Technology, Hellenic Open University, Athens, Greece. · School of Pedagogical and Technological Education (ASPETE), Athens, Greece.

Dynamical Systems Chaotic Dynamical Systems Physics-Informed ML

Jul 24, 2026cond-mat.dis-nn

Multiplicity of Stable Attractors in Disordered Neural Models

We show how large-deviation statistics allows one to obtain reliable estimates of the multiplicity of stable fixed-points in a model of neural ordinary differential equations previously employed in computational tasks. The result is obtained by developing a suitable perturbative method in the amplitude of the disorder. It turns out that for not-too-large coupling strengths there are no qualitative differences between the symmetric case, when the dynamics is a purely gradient evolution, and the asymmetric case, when limit cycles and chaos can, in principle, arise. The selection of this specific model is dictated by pedagogical reasons, but we are confident that the approach can be extended to other many-degree-of-freedom dynamical models characterized by different classes of random coupling matrices.

Raffaele Marino, Roberto Livi, Antonio Politi

Butterfly Decisions srl, Via dei Principati 74 - 84122 Salerno, Italy · University of Florence, Department of Physics and Astronomy, Via G. Sansone 1 - 50019 Sesto Fiorentino (FI), Italy and Istituto dei Sistemi Complessi, CNR, Via Madonna del Piano 10 – 50019 Sesto Fiorentino (FI), Italy · Istituto dei Sistemi Complessi, CNR, Via Madonna del Piano 10 – 50019 Sesto Fiorentino (FI), Italy and Institute of Pure and Applied Mathematics, Department of Physics, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

Dynamical Systems Neural ODEs

Jul 24, 2026quant-ph

QC-PHAST Search: Classical--Quantum Query Benchmarks for Finite-Pool Rare-Regime Discovery

Rare-regime discovery in parameterized dynamical systems is an active-search problem: find one verified parameter at which a scientifically defined qualitative threshold is crossed, even when acceptable candidates are rare, nonconvex, or fragmented. We introduce Quantum-Classical Phase-space and Stability-Threshold Search (QC-PHAST), an evidence-gated decision protocol and query-accounting framework for finite candidate libraries. A candidate induces a dynamical object, simulator-derived criticality score, and verified first-hit predicate. Scientific metadata and charged pilot evidence are used to assess whether equation-aware search, scalar-score active search, predicate-only search, or only a query-model comparison is admissible. The quantum row is the inherited Grover/Boyer--Brassard--Hoyer--Tapp (BBHT) unknown-

M

marked-set query reference; it is not a new quantum-search theorem, materialized circuit, or hardware-speedup claim. The result is a regime map. Direct boundary constructions, geometry controls, online simulator loops, and learned-label accounting further identify when classical structure, false positives, calibration cost, or state preparation erases the query-model margin. QC-PHAST is therefore an auditable protocol for deciding when a finite-pool marked-set reference is informative and when classical or resource-aware search should dominate.

Harsh Milind Tirhekar, Chandrajit Bajaj

Department of Computer Science College of Natural Sciences The University of Texas at Austin Austin, TX, USA · Department of Computer Science Oden Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, TX, USA

Dynamical Systems

Jul 23, 2026math.DS

Natural Invariant Measures for Chaotic Game Dynamics: Finding Order in Chaos

We study the long-term behavior of the Multiplicative Weights Update (MWU) algorithm in game settings where learning dynamics frequently fail to converge to Nash equilibria and instead exhibit Li-Yorke chaos. While such chaos precludes the prediction of specific long-term strategy profiles, it does not imply a lack of statistical structure. We demonstrate that natural invariant measures - a fundamental concept from ergodic theory - provide the rigorous framework necessary to find order within this chaos. Focusing on a two-strategy congestion game, we prove that these measures allow for a comprehensive statistical characterization of the dynamics. Crucially, we show that this framework extends beyond simple strategy frequencies to \emph{general observables}, enabling the precise calculation of long-term time averages for broad classes of economic metrics - including payoffs, social cost, and regret - despite chaos. Our results reveal that this simple learning algorithm captures the full spectrum of behaviors found in one-dimensional dynamical systems, from unique or multiple absolutely continuous invariant measures to complex periodic attractors as well as coexisting chaotic and stable (periodic) behaviors. By bridging game theory and dynamical systems, we show that statistical predictability is attainable even in the absence of pointwise convergence.

Jakub Bielawski, Thiparat Chotibut, Fryderyk Falniowski +2

Department of Mathematics, Krakow University of Economics, Rakowicka 27, 31-510 Kraków, Poland · Chula Intelligent and Complex Systems Lab, Department of Physics, Faculty of Science, Chulalongkorn University,Bangkok 10330, Thailand · Department of Mathematical Sciences, Indiana University Indianapolis,402 N. Blackford Street, Indianapolis, Indiana 46202, USA +1

Dynamical Systems Game Theory Chaotic Dynamical Systems

Jul 23, 2026eess.SY

Interaction Dynamics Modeling and Predictive Control for Safe Steerable Catheter--Tissue Interaction

Steerable catheters are the primary tool for cardiac electrophysiology (EP) procedures including radiofrequency ablation, where the tip must be positioned precisely at target tissue while maintaining controlled, stable contact. The central control problem is therefore not merely tip tracking and not merely force regulation; it is the regulation of \emph{catheter--tissue interaction dynamics}. The interaction state must encode how the tip moves relative to tissue, how persistent friction and contact forces bias that motion, and how safety limits reshape what motion is physically allowable. Existing methods regulate these interaction dynamics through different mechanisms. Classical impedance control~\cite{hogan1985} shapes the tip port as a virtual mechanical impedance

Z(s) = M_d s^2 + D_d s + K_d

, providing passive compliance without an explicit contact model. Three complementary design requirements motivate the present formulation: \textbf{(i)}~an explicit force-related constraint, \textbf{(ii)}~compensation for steady error under persistent loading, and \textbf{(iii)}~a prediction model that can incorporate trajectory and actuator information.

Yongyan Cao

Voryx Robotic LLC, San Jose, CA 95136, USA

Dynamical Systems Model Predictive Control Surgical Robotics