Dynamical Systems

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Overview

Dynamical systems describe how a system's state changes over time and explain behaviors such as equilibria, oscillations, attractors, bifurcations, and chaos. Researchers analyze these behaviors mathematically and learn models from data using approaches such as Koopman operators, neural ODEs, physics-informed networks, and structure-preserving Hamiltonian models. Applications range from forecasting Lorenz and pendulum dynamics to modeling reaction-diffusion processes, neural activity, and robot motion, often with control and safety guarantees. A central challenge is that accurate short-term predictions do not ensure faithful long-term behavior: learned models can drift, violate physical constraints, or fail to generalize to unseen regimes.

Latest papers

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482 papers56 papers in the last four weeks, up 19% on the four weeks before. 0.3% of all new papers.

Apr 18, 2026

Emergence Transformer: Dynamical Temporal Attention Matters

The Transformer, a breakthrough architecture in artificial intelligence, owes its success to the attention mechanism, which utilizes long-range interactions in sequential data, enabling the emergent coherence between large language models (LLMs) and data distributions. However, temporal attention, that is, different forms of long-range interactions in temporal sequences, has rarely been explored in emergence phenomenon of complex systems including oscillatory coherence in quantum, biophysical, or climate systems. Here, by designing dynamical temporal attention (DTA) with time-varying query, key, and value matrices, we propose an Emergence Transformer. This architecture allows each component to interact with its own or its neighbors' past states through dynamical attention kernels, thereby enabling the promotion and/or suppression of the emergent coherence of components. Interestingly, we uncover that neighbor-DTA consistently promotes oscillatory coherence, whereas self-DTA exhibits an optimal attention weight for coherence enhancement, owing to its non-monotonic dependence on network structure. Practically, we demonstrate how DTA reshapes social coherence, suggesting strategies to either enhance agreement or preserve plurality. We further apply DTA to the paradigmatic Hopfield neural network, achieving emergent continual learning without catastrophic forgetting. Together, these results lay a foundation and provide an immediate paradigm for modulating emergence phenomenon in networked dynamics only using DTA.

Zihan Zhou, Bo-Wei Qin, Kai Du +1

School of Mathematical Sciences and Shanghai Center for Mathematical Sciences, Fudan University, 200433 Shanghai, China · Research Institute of Intelligent Complex Systems, Fudan University, 200433 Shanghai, China · Shanghai Artificial Intelligence Laboratory, 200232 Shanghai, China +1

Dynamical SystemsTemporal Attention
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Apr 17, 2026

Spectral Kernel Dynamics for Planetary Surface Graphs: Distinction Dynamics and Topological Conservation

The spectral kernel field equation R[k] = T[k] lacks a conservation-law analog. We prove (i) the fixed-point flow is strictly volume-expanding (tr DF > 0), precluding automatic conservation, and (ii) the conservation deficit per mode equals the Hessian stability margin exactly: D_m = -Delta'. Closing the deficit requires a scene-side compensating contribution, which we formalise as the distinction dynamics equation dc/dt = G[c, h_t], with MaxCal-optimal realisation G_opt. On fixed-topology 3D surface graphs we derive a conditional topology-preserving compression theorem: retaining k >= beta_0 + beta_1 modes (under a spectral-ordering assumption) preserves all Betti-number charges; we include a worked short-cycle counterexample (figure-eight) calibrating when the assumption fails. A triple necessary spectral diagnostic -- Fiedler-mode concentration, elevated curl energy, anomalous beta_1 -- is derived for planetary drainage networks at O(N) cost. Two internal real-data sequences serve as preliminary consistency checks; full benchmarks and adaptive-topology extensions are deferred.

Jnaneshwar Das

School of Earth and Space Exploration, Arizona State University

Dynamical SystemsSpectral MethodsTopological Data Analysis
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Apr 16, 2026

Stability and Generalization in Looped Transformers

Looped transformers promise test-time compute scaling by spending more iterations on harder problems, but it remains unclear which architectural choices let them extrapolate to harder problems at test time rather than memorize training-specific solutions. We introduce a fixed-point based framework for analyzing looped architectures along three axes of stability -- reachability, input-dependence, and geometry -- and use it to characterize when fixed-point iteration yields meaningful predictions. Theoretically, we prove that looped networks without recall have countable fixed points and cannot achieve strong input-dependence at any spectral regime, while recall combined with outer normalization reliably produces a regime in which fixed points are simultaneously reachable, locally smooth in the input, and supported by stable backpropagation. Empirically, we train single-layer looped transformers on chess, sudoku, and prefix-sums and find that downstream performance tracks the framework's predictions across tasks and architectural configurations. We additionally introduce internal recall, a novel recall placement variant, and show that it becomes competitive with -- and on sudoku, substantially better than -- standard recall placement once outer normalization is applied.

Asher Labovich

Neural Network GeneralizationDynamical SystemsNeural Network StabilityRecurrent Transformers
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Apr 16, 2026

A Nonlinear Separation Principle via Contraction Theory: Applications to Neural Networks, Control, and Learning

This paper establishes a nonlinear separation principle based on contraction theory and derives sharp stability conditions for recurrent neural networks (RNNs). First, we introduce a nonlinear separation principle that guarantees global exponential stability for the interconnection of a contracting state-feedback controller and a contracting observer, alongside parametric extensions for robustness and equilibrium tracking. Second, we derive sharp linear matrix inequality (LMI) conditions that guarantee the contractivity of both firing rate and Hopfield neural network architectures. We establish structural relationships among these certificates-demonstrating that continuous-time models with monotone non-decreasing activations maximize the admissible weight space-and extend these stability guarantees to interconnected systems and Graph RNNs. Third, we combine our separation principle and LMI framework to solve the output reference tracking problem for RNN-modeled plants. We provide LMI synthesis methods for feedback controllers and observers, and rigorously design a low-gain integral controller to eliminate steady-state error. Finally, we derive an exact, unconstrained algebraic parameterization of our contraction LMIs to design highly expressive implicit neural networks, achieving competitive accuracy and parameter efficiency on standard image classification benchmarks.

Anand Gokhale, Anton V. Proskurnikov, Yu Kawano +1

Center for Control, Dynamical Systems, and Computation, UC Santa Barbara, Santa Barbara, CA 93106 USA · Department of Electronics and Telecommunications, Politecnico di Torino, Italy, 10129 · Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

Dynamical SystemsContraction TheoryRecurrent Neural NetworksNeural Network StabilityNeural Network Control
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Apr 16, 2026

One-shot learning for the complex dynamical behaviors of weakly nonlinear forced oscillators

Extrapolative prediction of complex nonlinear dynamics remains a central challenge in engineering. This study proposes a one-shot learning method to identify global frequency-response curves from a single excitation time history by learning governing equations. We introduce MEv-SINDy (Multi-frequency Evolutionary Sparse Identification of Nonlinear Dynamics) to infer the governing equations of non-autonomous and multi-frequency systems. The methodology leverages the Generalized Harmonic Balance (GHB) method to decompose complex forced responses into a set of slow-varying evolution equations. We validated the capabilities of MEv-SINDy on two critical Micro-Electro-Mechanical Systems (MEMS). These applications include a nonlinear beam resonator and a MEMS micromirror. Our results show that the model trained on a single point accurately predicts softening/hardening effects and jump phenomena across a wide range of excitation levels. This approach significantly reduces the data acquisition burden for the characterization and design of nonlinear microsystems.

Teng Ma, Luca Rosafalco, Wei Cui +2

Dynamical SystemsSparse Identification of Nonlinear DynamicsNonlinear System IdentificationFew-Shot Learning
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Apr 14, 2026

On Common Misconceptions in Classical Vehicle Dynamics

Classical vehicle dynamics contains several widely adopted misconceptions that, while intuitively appealing, may lead to inconsistencies when examined under a rigorous mechanical framework. This paper revisits a number of such misconceptions, clarifying their domains of validity and highlighting potential sources of misunderstanding. The aim is not to dismiss established models, but to promote a more precise and scientifically grounded interpretation of key concepts.

Massimo Guiggiani

Department of Civil and Industrial Engineering, Universit`a di Pisa, Italy

Vehicle DynamicsDynamical Systems
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Apr 14, 2026

Stability and Geometry of Attractors in Neural Cellular Automata

Throughout the literature on Neural Cellular Automata (NCAs), it is often taken for granted that the systems learn attractors. This is shown through evolving the system for many timesteps and noting visual similarity to the goal state. There remain many questions after such an analysis. Namely, what kind of attractors do we have? Is their behavior ordered or chaotic? Can we estimate stability over very long time horizons? What really happens in the attractor when perturbations are applied? In this paper, we present a case study to help answer these questions, with methods drawn from the literature on dynamical systems theory. We use the growing gecko NCA of Mordvintsev et al. (2020) with deterministic cell updates as a case study. To the best of the authors' knowledge, we present the first visualizations of NCA attractor dynamics. We also analyze them using the Lyapunov and Fourier spectra, to reveal that the NCA displays oscillatory, periodic and quasi-periodic behavior, and that these behaviors arise early during training. This challenges the belief that NCAs learn fixed point attractors. Finally, we show that large perturbations to the attractor states can throw the NCAs into a secondary mode separate from the original attractor. We hope that this initial foray into NCA attractor dynamics expands the toolkit for NCA researchers to analyze the robustness and stability of their systems.

Mia-Katrin Kvalsund, James Stovold

Department of Physics, University of Oslo, Norway · Department of Computer Science, University of York, UK

Neural Cellular AutomataNeural Network RobustnessDynamical Systems
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Apr 14, 2026

Asymptotically Stable Gait Generation and Instantaneous Walkability Determination for Planar Almost Linear Biped with Knees

A class of planar bipedal robots with unique mechanical properties has been proposed, where all links are balanced around the hip joint, preventing natural swinging motion due to gravity. A common property of their equations of motion is that the inertia matrix is a constant matrix, there are no nonlinear velocity terms, and the gravity term contains simple nonlinear terms. By performing a Taylor expansion of the gravity term and making a linear approximation, it is easy to derive a linearized model, and calculations for future states or walkability determination can be performed instantaneously without the need for numerical integration. This paper extends the method to a planar biped robot model with knees. First, we derive the equations of motion, constraint conditions, and inelastic collisions for a planar 6-DOF biped robot, design its control system, and numerically generate a stable bipedal gait on a horizontal plane. Next, we reduce the equations of motion to a 3-DOF model, and derive a linearized model by approximating the gravity term as linear around the expansion point for the thigh frame angle. Through numerical simulations, we demonstrate that calculations for future states and walkability determination can be completed in negligible time. By applying control inputs to the obtained model, performing state-space realization, and then discretizing it, instantaneous walkability determination through iterative calculation becomes possible. Through detailed gait analysis, we discuss how the knee joint flexion angle and the expansion point affect the accuracy of the linear approximation, and the issues that arise when descending a small step.

Fumihiko Asano, Ning Lei, Taiki Sedoguchi

Graduate School of Advanced Science and Technology, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan

Robot LocomotionHumanoid Robot LocomotionGait AnalysisDynamical Systems
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Apr 13, 2026

Dynamic Modeling and Robust Gait Optimization of a Compliant Worm Robot

Worm-inspired robots provide an effective locomotion strategy for constrained environments by combining cyclic body deformation with alternating anchoring. For compliant robots, however, the interaction between deformable anchoring structures and the environment makes predictive modeling and deployable gait optimization challenging. This paper presents an experimentally grounded modeling and optimization framework for a compliant worm robot capable of traversing corrugated pipes. First, a hybrid dynamic locomotion model is derived, in which the robot motion is represented by continuous dynamics within a corrugation groove and discrete switching of anchoring positions between adjacent grooves. A slack-aware actuation model is further introduced to map the commanded gait input to the realized body-length change, and an energy model is developed based on physics and calibrated with empirical power measurement. Based on these models, a multi-objective gait optimization problem is formulated to maximize average speed while minimizing average power. To reduce the fragility of nominal boundary-seeking solutions, a kinematic robustness margin is introduced into the anchoring-transition conditions, leading to a margin-based robust gait optimization framework. Experimental results show that the proposed framework captures the dominant locomotion and energy-consumption behavior of the robot over the tested conditions, and enables robust gait optimization for achieving speed-power trade-off.

Xinyu Zhou, Yu Mei, Faith Thomson +3

Michigan State University, East Lansing, MI 48824 USA

Robot LocomotionEnergy-Efficient Robot LocomotionDynamical Systems
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Apr 13, 2026

Multi-Head Residual-Gated DeepONet for Coherent Nonlinear Wave Dynamics

Coherent nonlinear wave dynamics are often strongly shaped by a compact set of physically meaningful descriptors of the initial state. Traditional neural operators typically treat the input-output mapping as a largely black-box high-dimensional regression problem, without explicitly exploiting this structured physical context. Common feature-integration strategies usually rely on direct concatenation or FiLM-style affine modulation in hidden latent spaces. Here we introduce a different paradigm, loosely inspired by the complementary roles of state evolution and physically meaningful observables in quantum mechanics: the wave field is learned through a standard DeepONet state pathway, while compact physical descriptors follow a parallel conditioning pathway and act as residual modulation factors on the state prediction. Based on this idea, we develop a Multi-Head Residual-Gated DeepONet (MH-RG), which combines a pre-branch residual modulator, a branch residual gate, and a trunk residual gate with a low-rank multi-head mechanism to capture multiple complementary conditioned response patterns without prohibitive parameter growth. We evaluate the framework on representative benchmarks including highly nonlinear conservative wave dynamics and dissipative trapped dynamics and further perform detailed mechanistic analyses of the learned multi-head gating behavior. Compared with feature-augmented baselines, MH-RG DeepONet achieves consistently lower error while better preserving phase coherence and the fidelity of physically relevant dynamical quantities.

Zhiwei Fan, Yiming Pan, Daniel Coca

School of Engineering, Newcastle University, NE1 7RU, United Kingdom · School of Physical Science and Technology, ShanghaiTech University, Shanghai 200031, China

Deep Operator NetworksNeural OperatorsDynamical SystemsResidual Learning
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Apr 13, 2026

Learning to Test: Latent Representation Learning for Dynamic Instability Detection

Stability assessment is central to safety-critical systems governed by differential-algebraic equations (DAEs), whose dynamics are constrained by physical laws and admissibility conditions. These systems operate under stochastically varying environmental inputs, so stability must be reassessed as the context distribution shifts, yet repeated large-scale DAE simulation is prohibitive in high-dimensional or real-time settings. We propose a test-oriented learning framework that tests a trajectory-level chance constraint from deployment contexts alone. Rather than re-estimating physical parameters or re-solving the DAE, we learn a latent representation from baseline context-trajectory data that retains stability-relevant information and is regularized toward a tractable reference distribution, and calibrate a latent discrepancy test over the composite safety null. We establish exact validity for oracle calibration and asymptotic validity for its empirical implementation under class-conditional latent invariance, and show that the guarantee degrades gracefully when this invariance holds only approximately. On simulated dynamical systems and a power-system real-world example, our approach strikes a balance between validity and power, whereas other baseline methods each fail in at least one regime. An ablation that injects controlled violations of the invariance assumption shows that decisions stay correct while the violation is below the safety margin, as the theory predicts.

Minxing Zheng, Zewei Deng, Liyan Xie +1

Carnegie Mellon University · University of Minnesota

Dynamical SystemsPhysics-Informed MLOOD Detection
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Apr 12, 2026

Visualising the Attractor Landscape of Neural Cellular Automata

As Neural Cellular Automata (NCAs) are increasingly applied outside of the toy models in Artificial Life, there is a pressing need to understand how they behave and to build appropriate routes to interpret what they have learnt. By their very nature, the benefits of training NCAs are balanced with a lack of interpretability: we can engineer emergent behaviour, but have limited ability to understand what has been learnt. In this paper, we apply a variety of techniques to pry open the NCA black box and glean some understanding of what it has learnt to do. We apply techniques from manifold learning (principal components analysis and both dense and sparse autoencoders) along with techniques from topological data analysis (persistent homology) to capture the NCA's underlying behavioural manifold, with varying success. Results show that when analysis is performed at a macroscopic level (i.e. taking the entire NCA state as a single data point), the underlying manifold is often quite simple and can be captured and analysed quite well. When analysis is performed at a microscopic level (i.e. taking the state of individual cells as a single data point), the manifold is highly complex and more complicated techniques are required in order to make sense of it.

James Stovold, Mia-Katrin Kvalsund, Harald Michael Ludwig +2

University of York, UK · University of Oslo, Norway · Machine Learning Research Unit, TU Wien, Austria +2

Neural Cellular AutomataNeural Network InterpretabilityDynamical Systems
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Apr 12, 2026

Safety Guarantees in Zero-Shot Reinforcement Learning for Cascade Dynamical Systems

This paper considers the problem of zero-shot safety guarantees for cascade dynamical systems. These are systems where a subset of the states (the inner states) affects the dynamics of the remaining states (the outer states) but not vice-versa. We define safety as remaining on a set deemed safe for all times with high probability. We propose to train a safe RL policy on a reduced-order model, which ignores the dynamics of the inner states, but it treats it as an action that influences the outer state. Thus, reducing the complexity of the training. When deployed in the full system the trained policy is combined with a low-level controller whose task is to track the reference provided by the RL policy. Our main theoretical contribution is a bound on the safe probability in the full-order system. In particular, we establish the interplay between the probability of remaining safe after the zero-shot deployment and the quality of the tracking of the inner states. We validate our theoretical findings on a quadrotor navigation task, demonstrating that the preservation of the safety guarantees is tied to the bandwidth and tracking capabilities of the low-level controller.

Shima Rabiei, Sandipan Mishra, Santiago Paternain

The authors are with the Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute. · The author is with the Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute.

Zero-Shot RLSafe RLReinforcement LearningDynamical Systems
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Apr 10, 2026

On Feedback Speed Control for a Planar Tracking

This paper investigates a planar tracking problem between a leader and follower agent. We propose a novel feedback speed control law, paired with a constant bearing steering strategy, to maintain an abreast formation between the two agents. We prove that the proposed control yields asymptotic stability of the closed-loop system when the steering of the leader is known. For the case when the leader's steering is unavailable to the follower, we show that the system is still input-to-state stable with respect to the leader's steering viewed as an input. Furthermore, we demonstrate that if the leader's steering is periodic, the follower will asymptotically converge to a periodic orbit with the same period. We validate these results through numerical simulations and experimental implementations on mobile robots. Finally, we demonstrate the scalability of the proposed approach by extending the two-agent control law to an N-agent chain network, illustrating its implications for directional information propagation in biological and engineered flocks.

Xincheng Li, Tengyue Liu, Udit Halder

Department of Mechanical and Aerospace Engineering, University of South Florida

Input-to-State StabilityMulti-Robot Formation ControlFeedback ControlDynamical Systems
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Apr 9, 2026

Approximation of the Basset force in the Maxey-Riley-Gatignol equations via universal differential equations

The Maxey-Riley-Gatignol equations (MaRGE) model the motion of spherical inertial particles in a fluid. They contain the Basset force, an integral term which models history effects due to the formation of wakes and boundary layer effects. This causes the force that acts on a particle to depend on its past trajectory and complicates the numerical solution of MaRGE. Therefore, the Basset force is often neglected, despite substantial evidence that it has both quantitative and qualitative impact on the movement patterns of modelled particles. Using the concept of universal differential equations, we propose an approximation of the history term via neural networks which approximates MaRGE by a system of ordinary differential equations that can be solved with standard numerical solvers like Runge-Kutta methods.

Finn Sommer, Vamika Rathi, Sebastian Goetschel +1

Chair Computational Mathematics, Institute of Mathematics, Hamburg University of Technology, 21073 Hamburg, Germany

Dynamical SystemsComputational Fluid Dynamics
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Apr 7, 2026

Robust Learning of Heterogeneous Dynamic Systems

Ordinary differential equations (ODEs) provide a powerful framework for modeling dynamic systems arising in a wide range of scientific domains. However, most existing ODE methods focus on a single system, and do not adequately address the problem of learning shared patterns from multiple heterogeneous dynamic systems. In this article, we propose a novel distributionally robust learning approach for modeling heterogeneous ODE systems. Specifically, we construct a robust dynamic system by maximizing a worst-case reward over an uncertainty class formed by convex combinations of the derivatives of trajectories. We show the resulting estimator admits an explicit weighted average representation, where the weights are obtained from a quadratic optimization that balances information across multiple data sources. We further develop a bi-level stabilization procedure to address potential instability in estimation. We establish rigorous theoretical guarantees for the proposed method, including consistency of the stabilized weights, error bound for robust trajectory estimation, and asymptotical validity of pointwise confidence interval. We demonstrate that the proposed method considerably improves the generalization performance compared to the alternative solutions through both extensive simulations and the analysis of an intracranial electroencephalogram data.

Shuoxun Xu, Zijian Guo, Brooke R. Staveland +2

University of California at Berkeley · Zhejiang University · University of California at San Francisco

Dynamical SystemsRobust EstimationSystem IdentificationDistributionally Robust Optimization
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Apr 6, 2026

Differentiable Invariant Sets for Hybrid Limit Cycles with Application to Legged Robots

For hybrid systems exhibiting periodic behavior, analyzing the invariant set containing the limit cycle is a natural way to study the robustness of the closed-loop system. However, computing these sets can be computationally expensive, especially when applied to contact-rich cyber-physical systems such as legged robots. In this work, we extend existing methods for overapproximating reachable sets of continuous systems using parametric embeddings to compute a forward-invariant set around the nominal trajectory of a simplified model of a bipedal robot. Our three-step approach (i) computes an overapproximating reachable set around the nominal continuous flow, (ii) catalogs intersections with the guard surface, and (iii) passes these intersections through the reset map. If the overapproximated reachable set after one step is a strict subset of the initial set, we formally verify a forward invariant set for this hybrid periodic orbit. We verify this condition on the bipedal walker model numerically using immrax, a JAX-based library for parametric reachable set computation, and use it within a bi-level optimization framework to design a tracking controller that maximizes the size of the invariant set.

Varun Madabushi, Akash Harapanahalli, Samuel Coogan +1

School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA.

Dynamical SystemsReachability AnalysisFormal VerificationLegged Robotics
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Apr 6, 2026

Finite-Step Invariant Sets for Hybrid Systems with Probabilistic Guarantees

Poincare return maps are a fundamental tool for analyzing periodic orbits in hybrid dynamical systems, including legged locomotion, power electronics, and other cyber-physical systems with switching behavior. The Poincare return map captures the evolution of the hybrid system on a guard surface, reducing the stability analysis of a periodic orbit to that of a discrete-time system. While linearization provides local stability information, assessing robustness to disturbances requires identifying invariant sets of the state space under the return dynamics. However, computing such invariant sets is computationally difficult, especially when system dynamics are only available through forward simulation. In this work, we propose an algorithmic framework leveraging sampling-based optimization to compute a finite-step invariant ellipsoid around a nominal periodic orbit using sampled evaluations of the return map. The resulting solution is accompanied by probabilistic guarantees on finite-step invariance satisfying a user-defined accuracy threshold. We demonstrate the approach on two low-dimensional systems and a compass-gait walking model.

Varun Madabushi, Elizabeth Dietrich, Hanna Krasowski +1

Georgia Institute of Technology · University of California, Berkeley

Dynamical Systems
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Apr 4, 2026

Lotka-Sharpe Neural Operators for Control of Population PDEs

Age-structured predator-prey integro-partial differential equations provide models of interacting populations in ecology, epidemiology, and biotechnology. A key challenge in feedback design for these systems is the scalar ζζ, defined implicitly by the Lotka-Sharpe nonlinear integral condition, as a mapping from fertility and mortality rates to ζζ. To solve this challenge with operator learning, we first prove that the Lotka-Sharpe operator is Lipschitz continuous, guaranteeing the existence of arbitrarily accurate neural operator approximations over a compact set of fertility and mortality functions. We then show that the resulting approximate feedback law preserves semi-global practical asymptotic stability under propagation of the operator approximation error through various other nonlinear operators, all the way through to the control input. In the numerical results, not only do we learn ``once-and-for-all'' the canonical Lotka-Sharpe (LS) operator, and thus make it available for future uses in control of other age-structured population interconnections, but we demonstrate the online usage of the neural LS operator under estimation of the fertility and mortality functions.

Miroslav Krstic, Iasson Karafyllis, Luke Bhan +1

University of California, San Diego, USA · National Technical University of Athens, Greece · KTH Royal Institute Technology, Stockholm, Sweden

Feedback ControlPDE Operator LearningNeural OperatorsDynamical Systems
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Apr 3, 2026

On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations

This work establishes a rigorous bridge between infinite-dimensional delay dynamics and finite-dimensional Koopman learning, with explicit and interpretable error guarantees. While Koopman analysis is well-developed for ordinary differential equations (ODEs) and partially for partial differential equations (PDEs), its extension to delay differential equations (DDEs) remains limited due to the infinite-dimensional phase space of DDEs. We propose a finite-dimensional Koopman approximation framework based on history discretization and a suitable reconstruction operator, enabling a tractable representation of the Koopman operator via kernel-based extended dynamic mode decomposition (kEDMD). Deterministic error bounds are derived for the learned predictor, decomposing the total error into contributions from history discretization, kernel interpolation, and data-driven regression. Additionally, we develop a kernel-based reconstruction method to recover discretized states from lifted Koopman coordinates, with provable guarantees. Numerical results demonstrate reliable prediction of nonlinear delay systems, with potential relevance to future control applications.

Santosh Mohan Rajkumar, Dibyasri Barman, Kumar Vikram Singh +1

Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH, USA. · Department of Chemical, Paper, and Biomedical Engineering at Miami University, Oxford, OH, USA. · Department of Mechanical and Manufacturing Engineering at Miami University, Oxford, OH, USA.

Extended Dynamic Mode DecompositionDynamical SystemsKoopman Operator LearningLatent Dynamics ModelingNonlinear System Identification
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Apr 2, 2026

A Dynamic Toolkit for Transmission Characteristics of Precision Reducers with Explicit Contact Geometry

Precision reducers couple contact geometry, bearing support, structural deformation, and loading history. This paper presents a dynamic toolkit connecting distributed local contacts to the complete mechanical reaction path. Work-conjugate maps transfer displacement, reaction, and tangent contributions between contacts and rigid or reduced coordinates, with explicit allocation of contact and body elasticity. An implicit generalized-alpha solution distinguishes trial evaluations from accepted history. Contact records then support performance protocols and configured geometric or constitutive feedback. The numerical studies focus on mechanical coupling and pressure recovery. An idealized annular housing retains physical interfaces while its structural coordinates are reduced. Two reductions with similar static errors have cross-port response errors of 24.975 and 0.312 percent over the same frequency band relative to a common parent model. In a shared-pin example, a 20 micrometer radial displacement of one wheel changes the load on a second, fixed wheel by approximately 75 N. Removing cross-station compliance removes this incremental transfer on the tested sleeve-seating branch. A double-wheel cycloidal assembly relates torsional branch response to aggregate contact-load variation and normalized pressure fields. The pressure maxima remain sensitive to resolution despite small discrete force residuals. Further formulations specify how motion and contact records support precision, vibration, heat, wear, and durability models with their required inputs. The framework separates model representation and numerical resolution while retaining common definitions of motion, force, and observation.

Jiacheng Miao, Chao Liu, Qiliang Wang +2

Intelligent Equipment Division, CRRC Qishuyan Institute Co., Ltd, Changzhou, 213011, China · School of Mechanical Engineering, Dalian Jiaotong University, Dalian, 116028, China · Research Institute, CRRC Qishuyan Institute Co., Ltd, Changzhou, 213011, China

Dynamical SystemsMultibody DynamicsReduced-Order Modeling
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Apr 1, 2026

Oscillator-Based Associative Memory with Exponential Capacity: Theory, Algorithms, and Hardware Implementation

Associative memory systems enable content-addressable storage and retrieval of patterns, a capability central to biological neural computation and artificial intelligence. Classical implementations such as Hopfield networks face fundamental limitations in memory capacity, scaling at most linearly with network size. We present an associative memory architecture based on Kuramoto oscillator networks with honeycomb topology in which memories are encoded as stable phase-locked configurations. The honeycomb network consists of multiple cycles that share nodes in a chain-like arrangement, creating a one-dimensional lattice of chained+loops. We prove that this architecture achieves exponential memory capacity: a network of NN oscillators can store (2⌈nc/4⌉−1)m(2\lceil n_c/4 \rceil - 1)^m distinct patterns, where mm honeycomb cycles each contain ncn_c oscillators. Moreover, we fully characterize all stable configurations and prove that each memory's basin of attraction maintains a guaranteed minimum size independent of network scale. Simulations using charge-density-wave (CDW) oscillators validate predicted phase-locking behavior, demonstrating practical realizability in neuromorphic hardware.

Arie Ogranovich, Taosha Guo, Arvind R. Venkatakrishnan +3

Department of Mechanical Engineering, University of California, Santa Barbara, Santa Barbara, 93106, USA · Department of Electrical Engineering and Computer Science, University of California, Irvine, Irvine, CA, 92697, USA

Kuramoto ModelAssociative MemoryOscillatory Neural NetworksDynamical SystemsNeuromorphic Computing
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Apr 1, 2026

Bias Inheritance in Neural-Symbolic Discovery of Constitutive Closures Under Function-Class Mismatch

We investigate the data-driven discovery of constitutive closures in nonlinear reaction-diffusion systems with known governing PDE structures. Our objective is to robustly recover diffusion and reaction laws from spatiotemporal observations while avoiding the common pitfall where low residuals or short-horizon predictions are conflated with physical recovery. We propose a three-stage neural-symbolic framework: (1) learning numerical surrogates under physical constraints using a noise-robust weak-form-driven objective; (2) compressing these surrogates into restricted interpretable symbolic families (e.g., polynomial, rational, and saturation forms); and (3) validating the symbolic closures through explicit forward re-simulation on unseen initial conditions. Extensive numerical experiments reveal two distinct regimes. Under matched-library settings, weak polynomial baselines behave as correctly specified reference estimators, showing that neural surrogates do not uniformly outperform classical bases. Conversely, under function-class mismatch, neural surrogates provide necessary flexibility and can be compressed into compact symbolic laws with minimal rollout degradation. However, we identify a critical "bias inheritance" mechanism where symbolic compression does not automatically repair constitutive bias. Across various observation regimes, the true error of the symbolic closure closely tracks that of the neural surrogate, yielding a bias inheritance ratio near one. These findings demonstrate that the primary bottleneck in neural-symbolic modeling lies in the initial numerical inverse problem rather than the subsequent symbolic compression. We underscore that constitutive claims must be rigorously supported by forward validation rather than residual minimization alone.

Hanbing Liang, Ze Tao, Fujun Liu

Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology, Changchun 130022, P.R. China

Reaction-Diffusion SystemsPDE DiscoveryNeurosymbolic LearningDynamical SystemsNeural Surrogate ModelingSymbolic Regression
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Apr 1, 2026

Integer-State Dynamics of Quantized Spiking Neural Networks for Efficient Hardware Acceleration

Spiking neural networks (SNNs) support energy-efficient machine intelligence because event-driven computation and sparse activity map naturally to low-power digital hardware. In practical implementations, however, membrane states, synaptic weights, and thresholds are represented with finite-precision integer arithmetic. Quantization, clipping, and overflow can therefore alter network dynamics, not just approximate a higher-precision model. This paper adopts an integer-state dynamical perspective, modeling a hardware-oriented SNN as a deterministic map on a bounded integer lattice. Under this view, recurrence, periodic orbits, and regime changes become intrinsic properties of the system. We introduce a lightweight update rule with integer-valued states and shift-based leakage, and demonstrate the approach through exploratory simulations with network sizes N = 30-130, connection densities 0.1-0.9, and bit widths 4/8/16 over T = 1000 steps. The results show bounded and recurrent temporal structure with strong quantization sensitivity. The observed regimes depend heavily on representation semantics and scaling choices. These findings suggest that numerical precision acts as a dynamical design variable and highlight integer-state analysis as a useful framework for hardware-aware SNN co-design, motivating future work on attractor analysis, precision-aware training, and FPGA/ASIC validation.

Lei Zhang

Faculty of Engineering and Applied Science, University of Regina

Spiking Neural NetworksNeuromorphic ComputingNeural Network QuantizationDynamical Systems
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Apr 1, 2026

Predicting Dynamics of Ultra-Large Complex Systems by Inferring Governing Equations

Predicting the behavior of ultra-large complex systems, from climate to biological and technological networks, is a central unsolved challenge. Existing approaches face a fundamental trade-off: equation discovery methods provide interpretability but fail to scale, while neural networks scale but operate as black boxes and often lose reliability over long times. Here, we introduce the Sparse Identification Graph Neural Network, a framework that overcome this divide by allowing to infer the governing equations of large networked systems from data. By defining symbolic discovery as edge-level information, SIGN decouples the scalability of sparse identification from network size, enabling efficient equation discovery even in large systems. SIGN allows to study networks with over 100,000 nodes while remaining robust to noise, sparse sampling, and missing data. Across diverse benchmark systems, including coupled chaotic oscillators, neural dynamics, and epidemic spreading, it recovers governing equations with high precision and sustains accurate long-term predictions. Applied to a data set of time series of temperature measurements in 71,987 sea surface positions, SIGN identifies a compact predictive network model and captures large-scale sea surface temperature conditions up to two years in advance. By enabling equation discovery at previously inaccessible scales, SIGN opens a path toward interpretable and reliable prediction of real-world complex systems.

Qi Shao, Duxin Chen, Jiawen Chen +5

School of Mathematics, Southeast University, Nanjing, 210096, P. R. China. · School of Electronic Engineering and Computer Science, Queen Mary University of London, London, E1 4NS, UK. · School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK. +4

Dynamical SystemsNonlinear System IdentificationSparse Identification of Nonlinear DynamicsSpatiotemporal ForecastingSpatiotemporal GNNs
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Apr 1, 2026

Activation Saturation and Floquet Spectrum Collapse in Neural ODEs

We prove that activation saturation imposes a structural dynamical limitation on autonomous Neural ODEs h˙=fθ(h)\dot{h}=f_θ(h) with saturating activations (tanh⁡\tanh, sigmoid, etc.): if qq hidden layers of the MLP fθf_θ satisfy ∣σ′∣≤δ|σ'|\leδ on a region~UU, the input Jacobian is attenuated as \normDfθ(x)≤C(U)\norm{Df_θ(x)}\le C(U) (for activations with sup⁡x∣σ′(x)∣≤1\sup_{x}|σ'(x)|\le 1, e.g.\ tanh⁡\tanh and sigmoid, this reduces to CWδqC_Wδ^q), forcing every Floquet (Lyapunov) exponen along any TT-periodic orbit γ⊂Uγ\subset U into the interval [−C(U),  C(U)][-C(U),\;C(U)]. This is a collapse of the Floquet spectrum: as saturation deepens (δ→0δ\to 0), all exponents are driven to zero, limiting both strong contraction and chaotic sensitivity. The obstruction is structural -- it constrains the learned vector field at inference time, independent of training quality. As a secondary contribution, for activations with σ′>0σ'>0, a saturation-weighted spectral factorisation yields a refined bound C~(U)≤C(U)\widetilde{C}(U)\le C(U) whose improvement is amplified exponentially in~TT at the flow level. All results are numerically illustrated on the Stuart--Landau oscillator; the bounds provide a theoretical explanation for the empirically observed failure of tanh⁡\tanh-NODEs on the Morris--Lecar neuron model.

Nikolaos M. Matzakos

Currently visiting the Department of Mathematics, Chair of Dynamics, Control, Machine Learning and Numerics, Friedrich-Alexander-Universität Erlangen–Nürnberg (FAU), 91058 Erlangen, Germany.

Lyapunov Exponent EstimationNeural ODEsDynamical SystemsNeural Network Activation Functions
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Mar 31, 2026

Hierarchical Motion Planning and Control under Unknown Nonlinear Dynamics via Predicted Reachability

Autonomous motion planning under unknown nonlinear dynamics requires learning system properties while navigating toward a target. In this work, we develop a hierarchical planning-control framework that enables online motion synthesis with limited prior system knowledge. The state space is partitioned into polytopes and approximates the unknown nonlinear system using a piecewise-affine (PWA) model. The local affine models are identified once the agent enters the corresponding polytopes. To reduce computational complexity, we introduce a non-uniform adaptive state space partition strategy that refines the partition only in task-relevant regions. The resulting PWA system is abstracted into a directed weighted graph, whose edge existence is incrementally verified using reach control theory and predictive reachability conditions. Certified edges are weighted using provable time-to-reach bounds, while uncertain edges are assigned information-theoretic weights to guide exploration. The graph is updated online as new data becomes available, and high-level planning is performed by graph search, while low-level affine feedback controllers are synthesized to execute the plan. Furthermore, the conditions of classical reach control theory are often difficult to satisfy in underactuated settings. We therefore introduce relaxed reachability conditions to extend the framework to such systems. Simulations demonstrate effective exploration-exploitation trade-offs with formal reachability guarantees.

Zhiquan Zhang, Melkior Ornik

University of Illinois Urbana-Champaign, IL, 61801 USA.

Motion PlanningDynamical Systems
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Mar 31, 2026

Set-Based Value Function Characterization and Neural Approximation of Stabilization Domains for Input-Constrained Discrete-Time Systems

Analyzing nonlinear systems with stabilizable controlled invariant sets (CISs) requires accurate estimation of their domains of stabilization (DOS) together with associated stabilizing controllers. Despite extensive research, estimating DOSs for general nonlinear systems remains challenging due to fundamental theoretical and computational limitations. In this paper, we propose a novel framework for estimating DOSs for controlled input-constrained discrete-time systems. The DOS is characterized via newly introduced value functions defined on metric spaces of compact sets. We establish the fundamental properties of these value functions and derive the associated Bellman-type (Zubov-type) functional equations. Building on this characterization, we develop a physics-informed neural network (NN) framework that learns the value functions by embedding the derived functional equations directly into the training process. The proposed methodology is demonstrated through two numerical examples, illustrating its ability to accurately estimate DOSs and synthesize stabilizing controllers from the learned value functions.

Mohamed Serry, S. Sivaranjani, Jun Liu

Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada. · Department of Mechanical and Mechatronics Engineering at the University of Waterloo. · School of Industrial Engineering, Purdue University, 15 N Grant Street, West Lafayette, 47907, Indiana.

Dynamical SystemsValue Function Estimation
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Mar 31, 2026

Hybrid Energy-Based Models for Physical AI: Provably Stable Identification of Port-Hamiltonian Dynamics

Energy-based models (EBMs) implement inference as gradient descent on a learned Lyapunov function, yielding interpretable, structure-preserving alternatives to black-box neural ODEs and aligning naturally with physical AI. Yet their use in system identification remains limited, and existing architectures lack formal stability guarantees that globally preclude unstable modes. We address this gap by introducing an EBM framework for system identification with stable, dissipative, absorbing invariant dynamics. Unlike classical global Lyapunov stability, absorbing invariance expands the class of stability-preserving architectures, enabling more flexible and expressive EBMs. We extend EBM theory to nonsmooth activations by establishing negative energy dissipation via Clarke derivatives and deriving new conditions for radial unboundedness, exposing a stability-expressivity tradeoff in standard EBMs. To overcome this, we introduce a hybrid architecture with a dynamical visible layer and static hidden layers, prove absorbing invariance under mild assumptions, and show that these guarantees extend to port-Hamiltonian EBMs. Experiments on metric-deformed multi-well and ring systems validate the approach, showcasing how our hybrid EBM architecture combines expressivity with sound and provable safety guarantees by design.

Simone Betteti, Luca Laurenti

RIAS Lab, The Italian Institute of Artificial Intelligence for Industry, Turin, 10129, IT · Delft Center for Systems and Control, TU Delft, Delft, 2600 AA, The Netherlands

Energy-Based ModelsDynamical SystemsNonlinear System IdentificationLyapunov StabilityPort-Hamiltonian Systems
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Mar 31, 2026

HyperKKL: Learning KKL Observers for Non-Autonomous Nonlinear Systems via Hypernetwork-Based Input Conditioning

Kazantzis-Kravaris/Luenberger (KKL) observers are a class of state observers for nonlinear systems that rely on an injective map to transform the nonlinear dynamics into a stable quasi-linear latent space, from where the state estimate is obtained in the original coordinates via a left inverse of the transformation map. Current learning-based methods for these maps are designed exclusively for autonomous systems and do not generalize well to controlled or non-autonomous systems. In this paper, we propose two learning-based designs of neural KKL observers for non-autonomous systems whose dynamics are influenced by exogenous inputs. To this end, a hypernetwork-based framework (HyperKKLHyperKKL) is proposed with two input-conditioning strategies. First, an augmented observer approach (HyperKKLobsHyperKKL_{obs}) adds input-dependent corrections to the latent observer dynamics while retaining static transformation maps. Second, a dynamic observer approach (HyperKKLdynHyperKKL_{dyn}) employs a hypernetwork to generate encoder and decoder weights that are input-dependent, yielding time-varying transformation maps. We derive a theoretical worst-case bound on the state estimation error. Numerical evaluations on four nonlinear benchmark systems show that input conditioning yields consistent improvements in estimation accuracy over static autonomous maps, with an average symmetric mean absolute percentage error (SMAPE) reduction of 29% across all non-zero input regimes.

Yahia Salaheldin Shaaban, Abdelrahman Sayed Sayed, M. Umar B. Niazi +1

Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), Abu Dhabi, UAE. · Univ Gustave Eiffel, COSYS-ESTAS, F-59657 Villeneuve d’Ascq, France. · Department of Decision and Control Systems and Digital Futures, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden.

Dynamical Systems
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