Tapas Kumar Pa, Dibakar Ghosh
The Murali-Lakshmanan-Chua (MLC) circuit is a well-recognized prominent nonlinear, nonautonomous, and dissipative electronic circuit having a versatile chaotic nature. Unraveling the dynamical synergy responsible for the genesis of extreme events in nonlinear dynamical systems is a prolific and spellbinding research area. The present study unveils the dynamical exposition of emerging extreme events in the MLC circuit concerning two different events being defined in the system. The large expansion of the chaotic attractor following the PM intermittency route plays the crucial role as the precursor behind the emergence of extreme events in the system. Our main finding reveals the prevalence of a force field due to the presence of externally applied periodic force in the system that creates the dynamical synergy that compels the chaotic trajectory traversing in its phase space to be largely deviated from the residing space, and this large deviation shows the signature of extreme events. Apart from the force field explication, we explored another two dynamical aspects that also interpret the mechanism behind the genesis of extreme events as the large deflection of the chaotic trajectory in the system: the decomposition of the phase space in stable and unstable manifolds concerning slow-fast dynamics and using Floquet multipliers. These two different aspects of calculations of the stable and unstable manifolds explicate the large excursion of the chaotic trajectory as extreme events from two different perspectives. We also analyzed the rare occurrences of the extreme events statistically using extreme value theory: the threshold \textit{excess values} follow the generalized Pareto distribution, and the inter-extreme-spike-intervals follow the generalized extreme value distribution.
Karthik Duraisamy
Scientific machine learning is increasingly being spoken of as universal emulators for classical numerical solvers for multi-scale partial differential equations, but most apparent successes can be explained by facts that also define their limits. Many successful benchmarks live on low-dimensional solution manifolds where any competent reduced model will interpolate well. More fundamentally, neural surrogates systematically under-resolve high-frequency content due to spectral bias, and coarse-graining compounds this problem through irreversible information loss. In many multi-scale problems, no architecture or training procedure can fully recover what the coarse representation discards. Two simple examples are used to characterize spectral bias, coarse-graining and error accumulation. We discuss why medium-range weather prediction on reanalysis data sits in a favorable sweet spot and why this will not generalize to genuinely chaotic multi-scale scenarios. We identify domains where neural surrogates offer genuine value, propose further research on neural-classical hybrids, and call for better reporting standards.
Mahaveer Prasad, Ahana Chakraborty, Thomas Iadecola, Manas Kulkarni, J. H. Pixley, Sriram Ganeshan, Justin H. Wilson
Apr 21, 2026·quant-ph·PDF In classical dynamical systems, stochastic feedback can stabilize otherwise unstable periodic orbits, giving rise to distinct controlled and uncontrolled phases as the rate of control application is varied. In this work, we apply these control protocols in classical, semiclassical, and quantum regimes to the kicked top, a paradigmatic model of quantum chaos. The quantum kicked top, modeled as the dynamics of a spin-S object, naturally interpolates between these regimes with the spin size S acting as an effective Planck constant. We show that the dynamics of the kicked top in classical, semiclassical, and fully quantum limits can all be controlled using stochastic feedback protocols. Comparing the full quantum dynamics to a truncated Wigner approximation that captures quantum noise but neglects interference beyond the Ehrenfest time, we find that low-moment observables are largely accounted for semiclassically, while the remaining discrepancy in higher moments is consistent with contributions from interference and possibly nonlinearities in rare trajectories that explore the compact phase space. We also find rapid purification in the numerics studied for all rates of control considered, suggesting that control quenches the top's ability to encode a qubit of quantum information even in the uncontrolled phase.
Matthew Nicol, Manpreet Singh, Andrew Torok
We investigate the competition between two distinct mechanisms generating stable laws in deterministic dynamical systems: slow mixing of the system and heavy-tailed observables. For heavy-tailed observables on polynomially mixing billiards with cusps we show these two mechanisms interact and there is a transition, depending on the mixing exponent and the index of the heavy-tailed observable, such that the limit law is determined by either the observable or the dynamics. We prove stable limit laws for heavy-tailed observables of the form $φ(x)= d(x,x_0)^{-\frac{2}α}, 0< α< 2$, where $x_{0} \in \partial Q$ is a generic point on the dynamical system given by the collision map of a polynomially mixing billiard $(T, Q, μ)$ with cusps. The observable $φ$ has a tail of stable index $α$, i.e. $μ(|φ|>t) \sim t^{-α}$. The billiard systems we consider have a slow mixing rate so that suitably scaled Hölder observables on the billiard satisfy a stable law of index $1/γ$, with $γ$ a function of the flatness of the cusps. We establish stable limit laws satisfied by Birkhoff sums of $φ$ for the parameter range $γ\in (1/2,1)$, $α\in (0,2)$ ($α\not =1$) as a function of $γ$ and $α$. As an application, in the setting of intermittent maps, we extend the results of~\cite{CNT2025} to cover all parameter values of the map and the observable $φ(x)= d(x,x_0)^{-\frac{1}α}$ (which has stable index $α$ if $x_0\not =0$) in the regime $0< α< 2$, $0<γ<1$. We show if $x_0=0$, the indifferent fixed point, then the stable law has index $(\frac{1}α+γ)^{-1}$.
A. Schmaus, N. Marwan, N. Molkenthin
Trajectories of units moving on networks are relevant for nonlinear dynamical systems as diverse as polymers, ocean drifters, and human mobility. Although RQA is a well-researched tool with applications in many areas, it has rarely been used for spatial trajectories on networks. Here, we explore the use of RQA for paths on networks. We find that path dynamics on networks display recurrence patterns that are not often described in other applications of recurrence analysis. In particular, the combination of diagonal lines and perpendicular diagonal lines, indicates backtracking paths. We find that recurrence analysis for path dynamics on networks can be helpful to a) better understand the network structure if dynamic and recurrence plots are known, b) better understand the dynamics if network and recurrence plots are known, and c) understand the interaction between path dynamics and the underlying network.
Niraj Agarwal, Timothy A. Smith, Sergey Frolov, Laura C. Slivinski
Machine learning emulators have shown extraordinary skill in forecasting atmospheric states, and their application to global ocean dynamics offers similar promise. Here, we adapt the GraphCast architecture into a dedicated ocean-only emulator, driven by prescribed atmospheric conditions, for medium-range predictions. The emulator is trained on NOAA's UFS-Replay dataset. Using a 24 hour time step, single initial condition, and without using autoregressive training, we produce an emulator that provides skillful forecasts for 10-15 day lead times. We further demonstrate the use of Mahalanobis distance as loss that improves the forecast skill compared to the Mean Squared Error loss by explicitly accounting for the correlations between tendencies of the target variables. Using spatial correlation analysis of the forecasted fields, we also show that the proposed correlation-aware loss acts as a statistical-dynamical regularizer for the slow, correlated dynamics of the global oceans, offering a better background forecast for downstream tasks like data assimilation.
Urban Duh, Marko Žnidarič
Dynamical properties of classical chaotic systems, for instance relaxation, can be understood as emerging from the time evolution of initially smooth long-wavelength densities to ever finer short-wavelength densities with fractal structure. Whether there is any analogous fractality by which one could characterize quantum many-body chaos is not known. By studying the spectral properties of the truncated operator propagator, we provide such structures. Namely, we show that the slowest-decaying operators, i.e., the leading Ruelle-Pollicott eigenvectors, have a nontrivial fractal dimension quantifying their non-locality, visible also in the divergence of their condition numbers. Furthermore, we find that unitarity imposes a constraint, i.e., an (approximate) equality, between the temporal decay rate of local correlations and this spatial operator fractal dimension. With this insight, a scenario for many-body quantum chaos becomes clear: over time, local operators evolve towards increasingly non-local ones with a quantifiable fractal structure, thereby naturally leading to effective non-unitary relaxation on the subspace of local operators - a kind of many-body Kolmogorov cascade in the space of operators. Our predictions are demonstrated in various quantum circuits: the kicked Ising model, brickwall circuits with a random 2-qubit gate, and dual-unitary circuits, where our results are exact.
Patricia Rodriguez, Caracé Gutiérrez, Juan P. Tarigo, Cecilia Stari, Arturo C. Marti
We present an experimental study of the Duffing--Holmes oscillator with a double-well potential, implemented as an analog electronic circuit under periodic external forcing. By systematically varying the forcing amplitude and frequency, we characterize the full dynamical landscape of the system through bifurcation diagrams, Poincaré maps, and maximum Lyapunov exponent calculations. The observed phenomenology includes period-doubling routes to chaos, periodic windows with multistability, dynamical intermittency, and antiperiodic orbits in which the trajectory recovers the global symmetry of the double-well potential. These results are synthesized into a high-resolution two-dimensional phase diagram in parameter space. The close agreement between all experimental diagnostics validates the fidelity of the analog implementation and demonstrates that continuous-time hardware provides a powerful platform for the quantitative study of nonlinear dynamics, free from the discretization artifacts inherent to numerical simulation.
Srishty Aggarwal, Rohan Raha, Mayank Pathak, Banibrata Mukhopadhyay
Apr 17, 2026·astro-ph.HE·PDF The general relativistic magnetohydrodynamic (GRMHD) simulations are widely used to study accretion disk and jet dynamics around a black hole. Despite strong observational evidences for intrinsically nonlinear behavior, the interpretations of GRMHD simulation results, more precisely the underlying timeseries, have not been well-explored by nonlinear timeseries analysis. In this work, we characterize the jet and disk dynamics of different GRMHD simulated flows using the nonlinear timeseries analysis. As diagnostic tools, we consider Higuchi fractal dimension (HFD), Hurst Index (H) and spectral slope. We implement them for two model disk frameworks: magnetically arrested disk (MAD) and standard and normal evolution (SANE), across a range of black hole spins with the Kerr parameter spanning from -0.9375 to 0.9375. We simulate the disk/jet systems by two well-documented codes: HARMPI and BHAC, and obtain, respectively, low and high temporally resolved timeseries data. For both jet and disk dynamics, MADs are characterized by higher HFD, lower H and flatter spectral slopes than SANEs. High HFD in MAD could be due to its intermittent variability and indicates that it has lesser long-range temporal correlations than SANE. Moreover, HFD in MAD decreases with spin magnitude owing to increase in collimated, hence ordered, jets. However, in SANE, it increases with spin for positive ones due to interplay of winds and jets. Extending our analysis to observations, we attempt to segregate the classes of black hole: GRS 1915+105, into MAD- and SANE-like clusters based on their spectral properties extracted from X-ray data. The mean HFD of MAD-like cluster is higher than SANE-like cluster, thus, corroborating with the simulation results. Our work highlights the role of nonlinear timeseries analysis to understand the underlying dynamics of accretion flows and their connection to magnetic regulation.
Álvaro G. López, Inés P. Mariño, Alfonso Delgado-Bonal
Dissipative structures are open dynamical systems that sustain coherent macroscopic organization by continuously exchanging energy and matter with their environment and generating entropy. A recent thermodynamic analysis of the paradigmatic Malkus--Lorenz waterwheel interpreted the Lorenz system as an engine, deriving an exact formula for its thermodynamic efficiency, and showing that efficiency tends to increase as the system is driven far from equilibrium while displaying sharp drops near the Hopf subcritical bifurcation to chaos. Here, we extend that single-engine framework to coupled dissipative structures. We introduce two canonical couplings -- master-slave coupling (series) and symmetric diffusive coupling (parallel) -- and prove two fundamental association laws allowing us to reduce the composite systems to an equivalent engine with a specified efficiency. We then apply these abstract results to coupled Lorenz waterwheels, deriving efficiency formulas consistent with the underlying power balance. We perform numerical simulations confirming that (a) series coupling induces an increase in thermodynamic efficiency, (b) parallel coupling averages the efficiency of engines and increases total energy flow, (c) synchronization is typically neutral or beneficial for efficiency except in narrow parameter regions, and (d) coupling modifies the curvature of entropy-generation trends. Our theorems suggest a mathematically rigorous and transparent route to define and compute thermodynamic efficiency for generalized flow networks, with potential application to complex systems energetics.
David Amaro-Alcalá, Carlos Pineda
Apr 16, 2026·quant-ph·PDF We demonstrate that the Ising all-to-all (ATA) model exhibits a range of dynamics, from integrable to chaotic, including mixed behaviour across symmetry blocks within a single system. While other works have explored the dynamics of all-to-all systems by varying parameters, we analyse a fixed set of parameters and examine the dynamics within different blocks. In addition to investigating the dynamical properties, we show that the system remains resilient to noise when the norm of the Hamiltonian representing the noise is close to 1. Our results are presented by mapping each symmetry sector of the system to a kicked top (KT) and observing that KT parameters for each sector depend on its dimension. This system, similar to the Bunimovich billiard for classical chaos, provides a new platform for studying dynamics determined by the symmetry sector, advancing quantum chaos research.
Anusree M, Akhila Henry, Pramod P Nair
Convolutional neural networks (CNNs) often exhibit poor generalisation in limited training data scenarios due to overfitting and insufficient feature diversity. In this work, a simple and effective chaos-based feature transformation is proposed to enhance CNN performance without increasing model complexity. The method applies nonlinear transformations using logistic, skew tent, and sine maps to normalised feature vectors before the classification layer, thereby reshaping the feature space and improving class separability. The approach is evaluated on greyscale datasets (MNIST and Fashion-MNIST) and an RGB dataset (CIFAR-10) using CNN architectures of varying depth under limited data conditions. The results show consistent improvement over the standalone (SA) CNN across all datasets. Notably, a maximum performance gain of 5.43% is achieved on MNIST using the skew tent map with a 3-layer CNN at 40 samples per class. A higher gain of 9.11% is observed on Fashion-MNIST using the sine map with a 3-layer CNN at 50 samples per class. Additionally, a strong gain of 7.47% is obtained on CIFAR-10 using the skew tent map at 200 samples per class. The consistent improvements across different chaotic maps indicate that the performance gain is driven by the shared nonlinear and dynamical properties of chaotic systems. The proposed method is computationally efficient, requires no additional trainable parameters, and can be easily integrated into existing CNN architectures, making it a practical solution for data-scarce image classification tasks.
Daniel Waltner, Boris Gutkin
Apr 15, 2026·quant-ph·PDF Semiclassical methods have been applied very successfully to describe the nontrivial transition from the quantum to the classical regime in $\textit{single}$-particle or at least $\textit{few}$-particle systems. Challenges on the way to an extension to $\textit{many}$-body systems result from the exponential proliferation of the number of classical orbits in chaotic systems and the exponential growth of the quantum Hilbert-space dimension with the particle number. To circumvent these problems, we apply here our recently developed duality relation. Considering the kicked spin chain as example for a many-body system, we show how the duality relation can be used to extract the classical orbits from the quantum spectrum. For coupled cat maps, we analyze the spectral statistics of chaotic many-body systems and discuss the double limit of large semiclassical parameter and large particle number.
Ivan I. Shevchenko
The Melnikov-Arnold integrals (MA-integrals) is a well-known instrument used to measure the splitting of separatrices in Hamiltonian systems. In this article, we explore how calculation of MA-integrals can be used as well to estimate sizes of secondary resonances. Within the standard map model, we show how the newly developed MA-based procedure allows one to estimate the sizes of secondary resonances of any order (up to the order of the optimal normal form), without relying on the cumbersome traditional normalization procedure.
Yangshuo Zhou, Jiao Wang
Apr 15, 2026·quant-ph·PDF We investigate a many-body interacting system of quantum kicked rotors, where each rotor resides in its respective quantum resonance. Rich many-body dynamics are found to emerge from the interplay between the principal and secondary resonances. In particular, for both the wavepacket and bipartite entanglement entropy, we analytically demonstrate three distinct dynamical regimes -- quadratic spreading (growth), period-2 oscillation, and their hybrid -- governed by the respective symmetries of the relevant potentials. Based on these symmetries, the connection between the wavepacket and the entanglement dynamics is illustrated. Other related issues are also discussed, including higher-order resonance effects, the robustness of the predicted dynamical behaviors, extension to many-body kicked tops, and relevance to experimental studies.
Omid Bateniparvar, Farzan Farahmand, Ranajay Ghosh
Overlapping fish-scale architectures are among nature's most distinctive surface adaptations, combining protection, contact regulation, hydrodynamics, optical and directional mechanical response within a thin textured integument. Here, we show that their biomimetic structural analogues can host deterministic chaos. Biomimetic scale substrates develop chaotic flexural vibrations at modest amplitudes because bending activates unilateral contact and progressive jamming, while built-in asymmetry from unequal texturing biases the restoring response and shifts the onset of chaos. From continuum mechanics, we derive a singular reduced-order model (sROM) that reduces the scale-covered beam to a nonlinear oscillator whose parameters map directly to overlap, scale inclination, damping, forcing, and substrate stiffness. Finite element (FE) simulations validate the model in quasi-static bending and long-time forced response. Stroboscopic regime maps reveal a period-doubling cascade from period-1 to period-2 and period-4, ultimately chaos. Overlap and inclination determine the strength of post-engagement nonlinearity, whereas damping bounds the chaotic operating window. Unequal top-bottom scale distributions break the antisymmetry of the restoring response, generating offset force-displacement laws. This reduced symmetry does not accelerate instability; instead, it delays the onset of chaos and fragments the response into intermittent periodic windows, whereas restoring symmetry can paradoxically widen the chaotic regime. When the texture is sufficiently sparse or steep on one side, it remains dynamically inactive, and the beam behaves as a fully asymmetric one-sided system. The results identify biomimetic scale substrates as a distinct class of contact-rich architectured metasurfaces in which chaos is programmable through geometry rather than large deflection or constitutive nonlinearity.
Marcel Novaes
Apr 14, 2026·quant-ph·PDF We discuss the semiclassical approximation to transport problems in quantum chaotic systems. The figures of merit are moments of the transmission matrix and of the time delay matrix. After reviewing a few results obtained by treating these matrices are random matrices, we show how expressions for their elements in terms of sums over trajectories lead to diagrammatic formulations that correspond to perturbative calculations. This semiclassical approach agrees with random matrix theory when it should, and allows further elements to be incorporated, like tunnel barriers, superconductors, absorption effects. We also discuss how this approach can be encoded in matrix integrals, resulting in a powerful and versatile theory that is amenable to algebraic solutions.
Barbara Dietz
Neutrino billiards serve as a model system for the study of aspects of relativistic quantum chaos. These are relativistic quantum billiards consisting of a spin-1/2 particle which is confined to a planar domain by imposing boundary conditions on the spinor components which were proposed in [Berry and Mondragon 1987, {\it Proc. R. Soc.} A {\bf 412} 53) . We review their general features and the properties of neutrino billiards with shapes of billiards with integrable dynamics. Furthermore, we review the features of two neutrino billiards with the shapes of billiards generating a chaotic dynamics, whose nonrelativistic counterpart exhibits particular properties. Finally we briefly discuss possible experimental realizations of relativistic quantium billiards based on graphene billiards, that is, finite size sheets of graphene.
Steven Tomsovic
Apr 14, 2026·quant-ph·PDF Through semiclassical methods the subject of quantum chaos motivates and depends on Hamiltonian chaos research. Presented here is a selection of Hamiltonian chaos topics that in this way get directly related to any of a variety of quantum chaos research problems. The chapter begins with a description of various useful theoretical and computational tools of chaos research, e.g.~surfaces of section, paradigms of chaos, stability analysis, and symbolic dynamics... This is followed by discussions regarding the geometry of chaos, how chaotic systems respond to perturbations, and the complexification of Hamiltonian dynamics. The emphasis is on intuitive explanations and illustrations of various ideas with the references containing more mathematically rigorous expositions.
Akira Shudo
In generic Hamiltonian systems that are neither completely integrable nor fully chaotic, phase space consists of a mixture of regular and chaotic components. In classical dynamics, transitions between different invariant sets in phase space are strictly forbidden, and these sets act as dynamical barriers to one another. In quantum mechanics, in contrast, wave effects allow transitions through such dynamical barriers. This process, known as dynamical tunneling, refers to penetration through dynamical barriers in phase space and was first recognized in the early 1980s. Since then, various aspects of dynamical tunneling have been elucidated, significantly advancing our understanding of such a novel quantum phenomenon. In this article, we provide an overview of several phenomenological perspectives of dynamical tunneling, including chaos-assisted and resonance-assisted tunneling, and also introduce approaches based on classical mechanics extended into the complex domain. In particular, we seek to clarify what is meant by the common claim that "chaos leads to an enhancement of the tunneling probability", which is often made when dynamical tunneling is dressed. We discuss what regime this refers to and, if such an enhancement occurs, what its likely origin is.