288 papers
Self-excited oscillations in multi-degree-of-freedom systems subjected to discontinuous forcing
This study investigates the existence and stability of limit cycles resulting from self-excited oscillations in linear multi-degree-of-freedom systems subjected to discontinuous, state-dependent forcing. Using the method of averaging and slow-flow phase-plane analysis, analytical expressions are derived for the amplitudes and stability boundaries of limit cycles in a two-degree-of-freedom system. The analysis demonstrates that stable limit cycles may exist in all natural modes, with the steady-state response governed by initial conditions in regimes of multistability. A central contribution of this work is the identification and analytical characterization of the stability-axis-flipping (SAF) bifurcation, which serves as the governing mechanism for the exchange of stability between modes. The framework is then systematically extended to systems with higher degrees of freedom, confirming that the SAF bifurcation remains a universal feature, even under varying feedback configurations. The steady-state dynamics, summarized through stability maps and validated by numerical simulations, delineate the existence and stability regions of modal limit cycles as functions of key system parameters. These results provide efficient criteria for guiding optimization studies to mitigate or generate limit cycles at targeted frequencies in flexible mechanical structures.
2604.02940Apr 2026
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Phase-space organization of the elastic pendulum: chaotic fraction, energy exchanges, and the order-chaos-order transition
We study the phase-space organization of the planar elastic pendulum as a function of its two dimensionless control parameters: the reduced energy $R$ and the squared frequency ratio $μ$. By randomly sampling the isoenergetic volume to classify trajectories as oscillatory, rotational, or chaotic across the $(μ, R)$ parameter plane, we obtain a global portrait of the coexistence and competition between dynamical regimes. The chaotic fraction is not uniformly distributed across the parameter plane but concentrates in a well-defined central cloud whose ridge follows a linear relation in the $(μ, R)$ plane and whose maximum does not exceed $70\%$ of the available phase space. The order-chaos-order transition is not a global property of the parameter plane but occurs specifically in the central region surrounding this cloud: along paths that traverse it, oscillatory orbits progressively give way to chaotic trajectories, which in turn yield to rotational orbits as the energy grows, revealing a clear sequential mechanism underlying the transition. The onset of rotational motion is gradual rather than sharp, reflecting a strong dependence on initial conditions. By decomposing the total energy into spring-like, pendulum-like, and coupling contributions, we establish a direct correspondence between the coupling power and the abundance of chaotic trajectories, showing that enhanced inter-mode energy exchange is a reliable indicator of dynamical complexity. These results provide a comprehensive and quantitative map of the dynamical regimes of the elastic pendulum, clarifying the structure of the chaotic cloud and connecting it to the underlying mode-coupling mechanisms.
2604.01503Apr 2026
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Slow spectral dynamics of shot noise in the Kuramoto model: the role of microscopic regularity
Finite-size effects in the Kuramoto model are known to induce collective fluctuations even below the critical coupling, where the thermodynamic limit predicts complete asynchrony. While the shot-noise approach developed in our recent work accurately describes the power spectrum of these fluctuations for random frequency sampling, the present study reveals that the microscopic realization of the frequency distribution plays a crucial role. We show that a deterministic (quasi-uniform) selection of natural frequencies from the same Lorentzian distribution leads to qualitatively different dynamics: the shot noise spectrum exhibits anomalously slow oscillatory behavior, manifesting as wave-like patterns in time-frequency representations. The period of these oscillations scales linearly with the system size and matches the frequency spacing between neighboring oscillators near the distribution center. Numerical simulations confirm that these slow spectral dynamics arise from resonant interactions facilitated by the regular frequency structure, which are absent for random sampling. Our findings demonstrate that identical integral frequency distributions do not guarantee equivalent collective dynamics, highlighting the necessity of accounting for the fine structure of microscopic parameters in finite-size populations.
2603.28523Mar 2026
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Scrambling at the genesis of chaos
The presence of chaos in classical Hamiltonian systems is witnessed by its maximal Lyapunov exponent, that quantifies the instability of motion through the exponential growth of indicators such as the trace of the stability matrix or the out-of-time-ordered correlator. On the other hand, integrable dynamics near unstable fixed points, which are in turn characterized by a stability exponent, can also induce such exponential growth. Following the paradigm of integrability-breaking as driven by nonlinear resonances that hallmarks the genesis of chaos, the integrability-chaos transition is universally described by a periodic perturbation applied to a generic pendulum. Remarkably, this means that within the corresponding separatrix dynamics, which is an unavoidable a consequence of the resonance scenario, both instability exponents must play a role as both dynamical regimes coexist. We report here the universality of the transition from instability to Lyapunov exponents, thus completing the resonance scenario at the level of indicators based on exponential growth. To achieve this goal we obtain an analytical expression for the time evolution near separatrices, which enables us to derive an analytical expression for the exponent that characterises chaos and its transition from local instability to global chaos. We support our claim for the universality of this mechanism by studying two paradigmatic examples of the integrability-to-chaos transition, namely the kicked rotor and the driven pendulum.
2603.26480Mar 2026
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Nonequilibrium ensemble averages using nonlinear response relations
The transient time correlation function (TTCF) method is widely used in molecular fluids to compute non-equilibrium transport quantities, providing improved signal-to-noise ratios in ensemble averages without requiring prohibitively large sample sizes. In spite of its success in molecular and turbulent fluid systems, the method has not been systematically explored for more general non-equilibrium dynamical systems, including geophysical applications where the invariant measure is typically unknown. In this work, we present an analytical and numerical investigation of the TTCF method for computing nonlinear response functions in systems far from equilibrium. We discuss its relation to the spectral theory of stochastic systems, highlighting regimes where linear theory is insufficient and the advantages of TTCF. The aim of this work is to provide a framework for studying transient and steady-state responses using the TTCF approach in a broad class of nonequilibrium systems.
2603.26275Mar 2026
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Conservative dynamics in phase oscillator networks
The interaction between phase oscillators is conservative if the phase volume is conserved throughout the dynamics. We derive a general condition, based on the notion of a pair-Hamiltonian, for the pairwise couplings to be conservative. The conservative networks with Winfree-type and Kuramoto-Daido-type couplings are also discussed. It is demonstrated that although, in contradistinction to genuine Hamiltonian dynamics, there is no exact pairwise symmetry of the Lyapunov exponents, the Lyapunov spectrum for a large network is nearly symmetric. The concept is also generalized to triplet and quadruplet couplings.
2603.25431Mar 2026
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Parametric Modulation of Nonlinear Coupling in the Hénon Heiles System: Resonances, Chaos, and Stabilization
We investigate parametric modulation of the nonlinear coupling in the Henon Heiles system, which directly modifies intrinsic resonance structure in a manner complementary to additive forcing. Canonical perturbation theory in extended phase space yields normal forms predicting resonance tongues scaling as $\sqrt{\varepsilon}$ near commensurate frequencies. Melnikov analysis quantifies separatrix splitting and chaos onset, confirmed by symplectic simulations showing transition from localized resonances to global transport via overlap. High-frequency averaging reveals potential stiffening that suppresses chaos. parametric modulation of nonlinear coupling provides an alternative route for generating combination resonances and influencing chaotic dynamics
2603.22875Mar 2026
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Thermalization of Weakly Nonintegrable FPUT and Toda Dynamics: A Lyapunov Spectrum Perspective
We study the thermalization slowing down of Fermi-Past-Ulam-Tsingou (FPUT) chains and of Toda chains with nonintegrable boundaries. We focus on the transition from FPUT to harmonic chains, from FPUT to Toda chains with fixed boundaries, and from nonintegrable open boundary Toda to integrable fixed boundary Toda. We compute the Lyapunov spectrum and analyze its scaling properties upon approaching integrable limits. We analyze the scaling of the largest Laypunov exponent, the rescaled Lyapunov spectrum, and the Kolmogorov-Sinai entropy. Using additional analytic arguments we demonstrate evidence that all three cases are operating in the regime of a Long Range Network of nonintegrable perturbations.
2603.22806Mar 2026
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Stochastic Web Map: Survival probability and escape frequency
We study transport and escape in the Stochastic Web Map (SWM), an area-preserving system with phase-space structure controlled by a symmetry parameter $q$ and nonlinearity $K$. By analyzing the survival probability $P_{\text{S}}(n)$ and escape frequency $P_{\text{E}}(\ln n)$, we show that in the chaotic regime escape dynamics is governed by a single time scale $n_{\text{typ}}\propto K^{-2}h^{2}$; here $h$ is the size of the escape horizon. Deviations at large $K$ and small $h$ indicate a breakdown of the quasilinear approximation. Then, upon rescaling the time by $n_{\text{typ}}$, escape statistics becomes universal, independent of $q$. These results demonstrate that escape is controlled by global transport rather than symmetry.
2603.20888Mar 2026
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Hierarchical fragmentation of regular islands in a discontinuous nontwist map
The destruction of regular regions in two-dimensional, area-preserving maps is traditionally described in terms of the breakup of invariant curves and the persistence of transport barriers. Here, we investigate how this scenario changes when continuity is lost. We study the extended standard nontwist map with a perturbation whose period differs from a full revolution on the cylinder. In this setting, the map becomes discontinuous on this cylinder while remaining smooth on the real line. Using escape times, the smaller alignment index (SALI), Lyapunov exponents, and finite-time recurrence time entropy (RTE), we find that regular islands are not enclosed by a single invariant curve but instead undergo hierarchical fragmentation into smaller regular components connected by chaotic channels. We show that trajectories initialized near elliptic points exhibit long trapping followed by escape, ruling out the existence of a global transport barrier or a last invariant curve. We demonstrate that finite-time RTE exhibits broad, asymmetric distributions with a clear spatial organization, with low values near island centers and high values along chaotic channels, persisting at fine scales. We also find persistent partial barriers, where trajectories remain trapped for extremely long times before escaping. By restoring continuity in a modified formulation, we recover smooth invariant curves and eliminate fragmentation, demonstrating that the hierarchical structure originates from discontinuity rather than twist violation alone.
2603.17120Mar 2026
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Dicovering the emergent nonlinear dynamics of acoustically levitated cube clusters
The complex behavior of many natural and engineered systems emerges from the interaction of a small number of effective degrees of freedom. Discovering the physical basis of the interactions between these degrees of freedom directly from experimental observations has been a longstanding challenge, particularly with respect to predicting the long-time dynamics of dynamical systems with unknown equations of motion. Here, we introduce a data-driven approach that is able to produce a generative model for the long-time dynamical behavior of systems with a weakly attracting manifold. We apply this method to an experimental dynamical system with two degrees of freedom: acoustically levitated pairs of cube-shaped particles, which cluster by sharing a single edge. In the acoustic trap, the center-of-mass of the cube cluster oscillates vertically about the levitation plane, while also oscillating about their flexible hinge-like connection. Depending on their initial condition, the hinge dynamics evolve about three distinct nonlinear dynamical attractors persisting for hundreds of cycles. In order to capture the underlying physics, we develop a numerical fitting procedure and extract a minimal nonlinear dynamical model that captures both the long-time dynamics of the cluster as well as the convergence onto the dynamical steady state. This dynamical model uncovers the nonlinear, non-reciprocal coupling between the center-of-mass motion and the hinge degree of freedom that stabilizes the dynamical attractors, which we subsequently confirm by independent finite-element methods. Our results demonstrate a novel data-driven method for the discovery of nonlinear models with long-timescale stable predictions.
2603.15894Mar 2026
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Storage and selection of multiple chaotic attractors in minimal reservoir computers
Modern predictive modeling increasingly calls for a single learned dynamical substrate to operate across multiple regimes. From a dynamical-systems viewpoint, this capability decomposes into the storage of multiple attractors and the selection of the appropriate attractor in response to contextual cues. In reservoir computing (RC), multi-attractor learning has largely been pursued using large, randomly wired reservoirs, on the assumption that stochastic connectivity is required to generate sufficiently rich internal dynamics. At the same time, recent work shows that minimal deterministic reservoirs can match random designs for single-system chaotic forecasting. Under which conditions can minimal topologies learn multiple chaotic attractors? In this paper, we find that minimal architectures can successfully store multiple chaotic attractors. However, these same architectures struggle with task switching, in which the system must transition between attractors in response to external cues. We test storage and selection on all 28 unordered system pairs formed from eight three-dimensional chaotic systems. We do not observe a robust dependence of multi-attractor performance on reservoir topology. Over the ten topologies investigated, we find that no single one consistently outperforms the others for either storage or cue-dependent selection. Our results suggest that while minimal substrates possess the representational capacity to model coexisting attractors, they may lack the robust temporal memory required for cued transitions.
2603.15155Mar 2026
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Tipping resonance in a chaotically forced ice age model
Many physical systems are forced by external inputs, which can sometimes take the form of chaotic variation. A particular example is found in applications related to weather and climate, where chaotic variation is prevalent across various timescales. If the system in question has multiple attracting solutions for a given range of forcing, rate-induced tipping can be triggered by the chaotic forcing, with the difference in timescales between the forcing and the system acting as a `rate' parameter. In this paper, we explore the interplay between these two timescales in a low-order model of ice age dynamics. The model exhibits bistability between two equilibria in one region of the parameter space and between an equilibrium and a periodic orbit in another region. When chaotic variation of the parameters is allowed within these bistable regions, the solutions of the forced system undergo rate-induced tipping from one attractor to another. Simulations of the forced system show that the timescale of the chaotic forcing induces a resonance-like behaviour, with an optimal timescale at which the likelihood of rate-induced tipping is at its maximum. We combine basin instability theory, finite-time Lyapunov exponents, and linear resonance analysis under periodic forcing to explain this resonance effect.
2603.15043Mar 2026
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Hierarchy of extreme-event predictability in turbulence revealed by machine learning
Extreme-event predictability in turbulence is strongly state dependent, yet event-by-event predictability horizons are difficult to quantify without access to governing equations or costly perturbation ensembles. Here we train an autoregressive conditional diffusion model on direct numerical simulations of the two-dimensional Kolmogorov flow and use a CRPS-based skill score to define an event-wise predictability horizon. Enstrophy extremes exhibit a pronounced hierarchy: forecast skill persists from $\approx 1$ to $> 4$ Lyapunov times across events. Spectral filtering shows that these horizons are controlled predominantly by large-scale structures. Extremes are preceded by intense strain cores organizing quadrupolar vortex packets, whose lifetime sharply separates long- from short-horizon events. These results identify coherent-structure persistence as a governing mechanism for the predictability of turbulence extremes and provide a data-driven route to diagnose predictability limits from observations.
2603.13789Mar 2026
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An angular-momentum preserving dissipative model for the point-mass N -body problem
A simple mathematical model emulating energy dissipation due to tidal effects is proposed. In this model, forces acting between masses remove energy but preserve the total angular momentum of the system. We study the effect of such forces on a particular family of orbits, namely those corresponding to central configurations, and show that a specific dependence on the mutual distances between the bodies leads to homographic equations equivalent to those of the two-body problem with dissipation. We then describe in detail the topology of solutions of the dissipative two-body system via Poincaré compactification. Finally, we present equations averaged over Keplerian motion showing no influence of the dissipation on periapsis precession.
2603.12452Mar 2026
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Symmetry-driven layered dynamics in the Kuramoto-Sivashinsky equation
In this work, we uncover a layered organization of the state space in the Kuramoto-Sivashinsky equation with periodic boundary conditions, in which multiple invariant sets coexist at fixed system parameters and are selected by the initial condition. Within this framework, both chaotic attractors and periodic orbits (traveling waves) can be systematically generated by amplifying a single initial condition and parameterized by the initial energy. As the energy increases, the period of the periodic orbits decreases according to an inverse scaling law. In transitional parameter regions, periodic dynamics at low initial energy is found to coexist with strange attractors at higher energy levels, revealing a unique layered landscape governed by the viscosity and the initial condition. We conjecture that this behavior is linked to continuous spatial translational symmetry, which is reflected in the degeneracy of the neutral part of the Lyapunov spectrum.
2603.09811Mar 2026
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Non-Normal Route to Chaos
Deterministic chaos is commonly associated with spectral criticality: exponential sensitivity is expected when Jacobian eigenvalues exceed unity in parts of the attractor, producing the local expansion that offsets contraction elsewhere. We show that this paradigm is incomplete in dimensions d>1. We construct a bounded 3D dynamical system whose Jacobian is pointwise spectrally contracting, namely all instantaneous eigenvalues remain strictly inside the stability region, yet the system develops a positive maximal Lyapunov exponent and undergoes a transition to chaos as a non-normality index increases at fixed spectral radius. The mechanism relies on the repeated regeneration of transient non-normal amplification through endogenous switching that reinjects trajectories into amplifying non-orthogonal directions. Although demonstrated here for a discrete-time map, the mechanism is geometric and applies more broadly to deterministic dynamical systems. These results show that chaos can emerge without spectral criticality and identify non-normality as an independent route to deterministic chaos.
2603.08191Mar 2026
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Jacobian determinant as a deformation field in static billiards
We develop a deformation-based framework for analyzing static billiard systems through the Jacobian determinant computed in noncanonical angular coordinates. Although these systems are conservative, the determinant is not identically equal to unity, generating structured domains of local phase-space expansion and contraction. We show numerically that these domains balance globally, providing a geometric manifestation of area preservation in noncanonical variables. The curves defined by det J = 1 act as deformation boundaries that intersect unstable periodic points and correlate with invariant manifolds. We prove analytically that period-two orbits restore exact unit determinant under composition, while higher-period orbits exhibit angular modulation consistent with reversibility. The Jacobian determinant thus reveals an additional geometric layer in phase-space organization and offers a complementary perspective on conservative billiard dynamics.
2603.07767Mar 2026
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Chaotic Oscillator Networks for Classification Tasks
Chaotic oscillators have gained significant attention in the research community because of their ability to reproduce and investigate the complex dynamics of real-world phenomena. Recent advances in the design of chaotic oscillator ensembles have led to the development of efficient signal processing frameworks that surpass traditional approaches. However, scaling such systems remains challenging due to the significant increase of computational resources and issues with training convergence. This study advances the state of the art by addressing the problem of data processing with ensembles of nonlinear oscillators that can be scaled up. In our approach, the processing is achieved as an anticipated local resonance or echo in a group of coupled chaotic oscillators, driven by external data input. Local resonance is enabled by tuning the coupling terms between the oscillators, which are approximated using the traditional artificial neural network and adapted to match the input feature distributions. Training the framework entails training this neural network to capture the dynamics of the entire oscillator system. The framework is evaluated using synthetic data and demonstrates an accuracy in machine learning classification task, while patterns recognition and dynamic system identification are also presented here as an extension of the functionality that involves additional modifications. Additionally, the universality of this approach is demonstrated by tests with different connections configurations between the oscillators and their types. The main advantage of the proposed framework is that it avoids hand-crafting explicit coupling terms, which requires expert knowledge and does not scale for large problems. Leveraging standard machine learning components simplifies both training and deployment of oscillator networks, enabling gradient-based optimization.
2603.16909Mar 2026
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Breakdown of Linear Response in Uniformly Hyperbolic Systems with Hierarchical Structure
Linear response theory asserts that sufficiently small external biases produce currents proportional to the applied force and forms the theoretical foundation of nonequilibrium transport. Here we demonstrate that linear response can break down even in uniformly hyperbolic deterministic systems when hierarchical asymmetry is present. Using a minimal class of uniformly expanding chaotic maps with hierarchical multiscale structure, we show that progressively finer transport channels become dynamically active as the applied bias decreases. The resulting force current relation is monotone and exhibits a hierarchical, fractal-like organization of activation thresholds. As a consequence, the effective mobility diverges as F to 0, demonstrating breakdown of linear response despite strong chaos and uniform hyperbolicity. The effect arises from deterministic multiscale activation rather than intermittency, stochastic noise, or singular invariant measures. These results identify hierarchy as an independent deterministic mechanism for nonperturbative transport response and demonstrate that uniform hyperbolicity alone does not guarantee the validity of linear response.
2603.03885Mar 2026
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