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Nonlinear Sciences

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nlin.PS 2026-05-13 2 theorems

Modulated oscillations persist across SNIPER bifurcations

Nonuniform relaxation oscillations near SNIPER bifurcations

Long-wavelength instabilities turn uniform relaxation orbits into spatially varying states that continue on both sides of the point in media

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Properties of spatially dependent relaxation oscillations near a SNIPER bifurcation are described. A SNIPER bifurcation creates a large-amplitude long-period periodic orbit via the annihilation of a pair of fixed points in a saddle-node bifurcation. We show that in spatially extended media, this orbit may undergo a long-wavelength instability, leading to spatially modulated oscillations that persist on both sides of the SNIPER. The oscillations take different forms depending on the system: a chimera state in a theta-reaction-diffusion model, and chaotic spiking in an activator-inhibitor-substrate model. The results are expected to have applications in a number of physical systems exhibiting SNIPER bifurcations, ranging from models of the nervous system through chemical reactions to nonlinear optics.
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nlin.CD 2026-05-13 2 theorems

Chaotic billiards stay uniform under noise

Stochastically perturbed billiards: fingerprints of chaos and universality classes

Weak random reflections leave chaotic tables ergodic while mapping integrable ones to the Evans model with varying boundary density.

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Billiards tables - a minimal model for particles moving in a confined region - are known to present classical (and quantum) different features according to their shape, ranging from strongly chaotic to integrable dynamics. Here we consider the role of a stochastic perturbation of the elastic reflection law, and show that while chaotic billiards maintain their key statistical feature, the behaviour for integrable billiard tables is completely different: it can be linked, for tiny perturbations, to Evans stochastic billiard, where at each collision the reflected angle is a uniformly distributed stochastic variable on $(-\pi/2,\pi/2$). The resulting spatial stationary measure has peculiar aspects, like being typically non uniform along the boundary, differently from any chaotic billiard table.
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nlin.AO 2026-05-13 2 theorems

Asymmetric delay enlarges active Ο€ state in 1D swarmalators

The role of asymmetric time delay and its structure in 1D swarmalators

Increasing delay in self-interaction models grows the ordered active Ο€ region, unlike symmetric delays that boost unsteady states.

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Swarmalators are a class of coupled oscillators that simultaneously synchronize in both space and phase, providing a minimal model for systems ranging from biological microswimmers to robotic swarms. Time delay is ubiquitous in such systems, arising from finite signal propagation speeds and sensory processing lags, yet its structural form, whether symmetric or asymmetric, has received little attention. Here, we study a one-dimensional swarmalator model with asymmetric time delay, in which the delay enters only the self-interaction terms of the spatial and phase dynamics, breaking the symmetry assumed in prior work. We identify various collective states such as async, static phase wave, static {\pi}, and active {\pi}, and derive analytical stability boundaries for each as a function of the coupling parameters and delay. Our analysis reveals that the asymmetric delay structure fundamentally reshapes the collective phase diagram: in particular, for the asymmetric delay models, increasing the delay systematically expands the active {\pi} state at the expense of other ordered states, in contrast to the symmetric delay model, which more strongly promotes the presence of unsteady states that are generally not well ordered. By providing closed-form stability conditions validated against numerical simulations, our work establishes that the internal structure of the delay, not merely its magnitude, is a decisive factor in determining the emergent collective behavior of swarmalator populations.
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nlin.PS 2026-05-12 2 theorems

Breathing cyclops states fill large parameter regions in oscillator networks

Breathing and Rotobreathing Cyclops States in Phase Oscillators with Inertia and Two-Harmonic Coupling

Nonstationary three-cluster configurations with inertia and dual harmonics emerge as central features rather than exceptions.

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Cyclops states - three-cluster configurations consisting of two synchronous groups and a solitary oscillator - dominate in ensembles of phase oscillators with inertia and multiple coupling harmonics [Phys. Rev. E 109, 054202 (2024)]. In this work, for the first time, we systematically study nonstationary cyclops states that preserve the three-cluster structure: breathing and rotobreathing cyclops states. We identify two scenarios for their destabilization: period doubling, leading to quasicyclops states while preserving frequency synchronization within the clusters, and the destruction of one or two clusters, resulting in the emergence of switching cyclops or multicluster states. We show that breathing and rotobreathing cyclops states occupy vast parameter regions of the second coupling harmonic and are key elements of the dynamics. The results are important for predicting and controlling complex collective states in ensembles with higher-order interaction harmonics of various natures.
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nlin.PS 2026-05-12 Recognition

Kagome lattice yields new coupled equations for localized waves

Asymptotic Analysis of discrete nonlinear localised modes in a Kagome lattice

Asymptotic analysis near a flat-band degeneracy produces a 2+1D system whose solitary-wave solutions match lattice simulations.

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We describe a nonlinear kagome lattice with nonlinear dynamics described by Klein-Gordon interactions with a scalar unknown at each node, such as might occur in a nonlinear electrical lattice. We show that the dispersion relation has three bands - a flat band and two other surfaces which may meet in Dirac points or be separated by a gap. By using multiple scales asymptotic methods, we find a variety of reductions to nonlinear Schrodinger (NLS) systems, some of which are similar to those obtained previously, and have the Townes soliton as a solution. We find a novel system of coupled NLS equations, by forming an asymptotic expansion for small amplitude weakly nonlinear waves around the point where the flat band meets the upper surface of the dispersion relation. We analyse this 2+1 dimensional system using Lie symmetries, and find further reductions to more complicated solitary wave solutions. Numerical simulations of the wave are also presented.
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nlin.PS 2026-05-12 2 theorems

Topological solitons model ball lightning as stable 3D structures

A Topological Soliton Model for Ball Lightning: Theory and Numerical Verification with the 3D Gross-Pitaevskii Equation

Simulations of the 3D Gross-Pitaevskii equation show conserved charge protects solitons at observed scales

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Ball lightning is one of the most mysterious atmospheric phenomena, whose long lifetime, penetrative ability, and stability are difficult to explain with traditional physical models. This paper proposes a novel theoretical framework, interpreting ball lightning as a projection of a high-dimensional topological soliton into three-dimensional space. Its essence is described by a nonlinear Schr\"odinger equation with attractive interaction, protected by a non-zero topological charge. Through numerical simulation of the three-dimensional Gross-Pitaevskii equation, we verify the core predictions of this model: in a Bose-Einstein condensate with attractive interactions, solitons carrying topological charge exhibit: (1)long-lived stability (topological charge conserved under perturbations); (2)low transmission probability (due to minimal overlap integral resulting from orthogonality with the ground state wavefunction); (3)energy and size scales consistent with actual observations. Theoretical analysis indicates that the soliton lifetime is governed by the system's decoherence rate, providing a natural explanation for the observed second-scale lifetimes. This work not only offers a self-consistent physical explanation for ball lightning but also provides a concrete scheme for the experimental preparation and observation of three-dimensional topological solitons.
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nlin.PS 2026-05-12 2 theorems

Simulations tie ball lightning to stable topological solitons

A Topological Soliton Model for Ball Lightning: Theory and Numerical Verification with the 3D Gross-Pitaevskii Equation

Charged objects in the 3D Gross-Pitaevskii equation live long, resist transmission, and match observed energies and sizes.

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Ball lightning is one of the most mysterious atmospheric phenomena, whose long lifetime, penetrative ability, and stability are difficult to explain with traditional physical models. This paper proposes a novel theoretical framework, interpreting ball lightning as a projection of a high-dimensional topological soliton into three-dimensional space. Its essence is described by a nonlinear Schr\"odinger equation with attractive interaction, protected by a non-zero topological charge. Through numerical simulation of the three-dimensional Gross-Pitaevskii equation, we verify the core predictions of this model: in a Bose-Einstein condensate with attractive interactions, solitons carrying topological charge exhibit: (1)long-lived stability (topological charge conserved under perturbations); (2)low transmission probability (due to minimal overlap integral resulting from orthogonality with the ground state wavefunction); (3)energy and size scales consistent with actual observations. Theoretical analysis indicates that the soliton lifetime is governed by the system's decoherence rate, providing a natural explanation for the observed second-scale lifetimes. This work not only offers a self-consistent physical explanation for ball lightning but also provides a concrete scheme for the experimental preparation and observation of three-dimensional topological solitons.
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nlin.PS 2026-05-11 Recognition

Fourier features and clustering sort chimera state types

Classification of Chimera States via Fourier Analysis and Unsupervised Learning

Method maps parameter regions and distinguishes coherent-incoherent patterns in networks of Rayleigh oscillators

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Chimera states are among the most intriguing phenomena in nonlinear dynamics, characterized by the coexistence of coherent and incoherent behavior in systems of coupled identical oscillators. Many methods have been proposed to detect chimera states and to distinguish their different types. However, such methods often suffer from important limitations that prevent sufficiently precise classification. In this work, we overcome the issue by considering a method based on Fourier analysis to determine key signal characteristics such as amplitude, phase, and frequency, jointly with an unsupervised clustering step acting on normalized total variations, measures of local spatial changes of the above-mentioned dynamical features. The proposed method allows us to identify regions in parameter space returning chimera states, but also to further distinguish between the different types. The method is applied to a network of Rayleigh oscillators, which has been shown to exhibit a rich variety of dynamical patterns.
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nlin.CD 2026-05-11 2 theorems

Resonant couplings recovered from noisy phase oscillator data

Reconstructing resonant phase oscillator interactions from noisy time series

Targeting normal form terms isolates leading drift dynamics and detects effective higher-order interactions despite observational noise.

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We present a method for reconstructing resonant interactions in weakly coupled phase oscillator systems from noisy time series. Instead of attempting to recover the full phase equations, which may be non-identifiable in the presence of bounded observational uncertainty, the method reconstructs the resonant normal form terms that determine the leading-order drift dynamics. We develop first-order and second-order reconstruction procedures based on finite libraries of resonant Fourier modes and least-squares estimation. We prove error bounds for the reconstructed coefficients under natural assumptions on the observation noise and the distribution of initial conditions. The second-order method detects effective resonant interactions generated by the interplay of nonresonant first-order couplings. Numerical examples illustrate the reconstruction of resonant subnetworks and emergent higher-order interactions.
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nlin.SI 2026-05-11 1 theorem

AKNS reductions create 23 shifted nonlocal NLS equations

Multi-place shifted nonlocal reductions of a multi-component AKNS system

One-soliton solutions from the Hirota method remain nonsingular for admissible parameter values.

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Starting from a multi-component AKNS system, we obtain new shifted nonlocal nonlinear Schr\"{o}dinger equations. We find 13 different shifted nonlocal nonlinear Schr\"{o}dinger equations with two-place nonlocalities and 10 shifted nonlocal nonlinear Schr\"{o}dinger equations with four-place nonlocalities. We first obtain one-soliton solutions of the multi-component AKNS system by the Hirota method. Applying the shifted nonlocal reduction formulas to this solution, we obtain one-soliton solutions for the shifted nonlocal nonlinear Schr\"{o}dinger equations. In cases yielding nontrivial solutions, we discuss the singularity structures of the solutions and show that the one-soliton solutions we obtain are nonsingular for certain values of the parameters. We plot representative nonsingular solutions obtained for admissible parameter values.
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nlin.AO 2026-05-11 2 theorems

LΓ©vy noise turns synchronized waves into random walks

Extreme forcing and wave dynamics in weakly nonlocally coupled excitable FitzHugh-Nagumo systems

Periodic forcing organizes regular traveling waves in coupled excitable units while heavy-tailed noise drives opposing waves that randomize.

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The influence of extreme external forcing on traveling-wave dynamics in an ensemble of weakly nonlocally coupled excitable FitzHugh--Nagumo systems is studied. Three types of external exposure are considered: periodic Gaussian pulses, periodic pulses modulated by Gaussian white noise, and L\'evy noise with tunable distribution parameters. Periodic forcing produces synchronization tongues with highly regular collective dynamics and may induce multiple traveling waves or coexistence of partial synchronization with wave propagation. In contrast, L\'evy noise suppresses regular behavior and generates a regime of counter-propagating waves, which with increasing intensity transitions to random walking dynamics. The study provides a comprehensive classification of the observed dynamical regimes and presents their organization in parameter space for different types of extreme external forcing.
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nlin.SI 2026-05-11 2 theorems

Toda lattice converges to KdV globally under scaling

The Korteweg-de Vries limit for the global dynamics of the Toda lattice

H^1 initial data yields all-time convergence to KdV via scaling, translation, and conserved quantities from integrability.

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It has been observed that the dynamics of the Toda lattice can be well described by solutions of the Korteweg-de Vries (KdV) equation in the continuum limit. We show that, under the KdV scaling and a suitable translation, the solution of the Toda lattice with H^1 initial data converges to that of the KdV equation globally in time. Our proof relies on tools from harmonic analysis and also on the construction and the conservation of mass and energy of the Toda lattice, the latter of which are derived from the completely integrable structure of the Toda lattice. As a consequence, we obtain long-wave KdV limits for the Toda lattice.
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nlin.AO 2026-05-11 2 theorems

Exact algebraic phases derived for three identical Kuramoto oscillators

Global Analytical Solution of the Identical Kuramoto Model for N=3 via Koopman Eigenfunctions

Koopman eigenfunctions convert the dynamics to time-dependent quartics whose roots are fixed by initial conditions.

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The Kuramoto model is a paradigmatic model of collective synchronization in coupled oscillator systems. Although its mathematical properties have been extensively investigated, exact phase trajectories from arbitrary initial conditions have been available only for the simplest case, N=2. In this study, we provide a global analytical solution for the phase trajectories of the all-to-all coupled Kuramoto model with identical oscillators for N=3. This solution is obtained by constructing Koopman eigenfunctions that relate the phases to time and reducing the phase dynamics to time-dependent quartic equations. The algebraic branch corresponding to the initial condition is then selected to recover the corresponding phase trajectory. This gives an explicit algebraic reconstruction of the nonlinear phase dynamics from Koopman eigenfunctions.
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nlin.PS 2026-05-08

The paper develops an analytical framework to compute the probability distribution of…

Soliton gas resolution and statistics of random wave fields in semiclassical integrable turbulence

A soliton gas resolution conjecture combined with a stochastic inverse scattering transform yields an explicit integral formula for the…

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We develop a general analytical framework for determining the probability distribution of random nonlinear wave fields governed by the focusing nonlinear Schr\"odinger equation (fNLSE) in regimes where typical realizations are dominated by solitons. We formulate the soliton gas resolution conjecture for the long-time evolution of slowly varying ("semiclassical") random initial states and implement a stochastic analogue of the inverse scattering transform by establishing a relationship between the spectral density of states of the underlying bound-state soliton gas and the probability density function (PDF) of the intensity of the resulting turbulent wave field. The derived explicit integral representation for the PDF is shown to be in excellent agreement with direct numerical simulations across several representative regimes of fNLSE integrable turbulence. The results have broad applicability to physical systems including water waves, nonlinear optics, and superfluids.
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nlin.SI 2026-05-08 2 theorems

Ernst equations split into universal trilinear kernel

The General Structure of Trilinear Equations

A tau-ratio form for the potential isolates a cubic trilinear sector that governs all second derivatives, shared across Tomimatsu-Sato cases

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We investigate trilinear structures as a natural extension of the Hirota bilinear formalism in integrable systems. While bilinear equations are associated with Grassmannian geometry and Pl\"ucker relations, trilinear equations suggest a higher algebraic structure involving three-slot couplings of tau functions. Focusing on the stationary axisymmetric Einstein equations, we show that when the Ernst potential is written in a tau-ratio form, the nonlinear equation decomposes into a cubic sector containing all second-derivative terms and a quartic gradient envelope. The cubic sector is identified with a YTSF-type trilinear kernel. We formulate a general trilinear kernel criterion and apply it to the Tomimatsu--Sato solutions. In particular, we demonstrate that the $\delta=3$ solution possesses the same trilinear kernel structure as the $\delta=2$ case, with a universal normalization up to a constant factor. These results suggest that the trilinear kernel represents a universal structure governing the highest-derivative sector of the Ernst system, providing a new perspective on integrability beyond the bilinear hierarchy.
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nlin.SI 2026-05-08

Tau-ratio form splits Ernst equation into trilinear kernel

The General Structure of Trilinear Equations

The cubic sector matches a YTSF-type structure for both Ξ΄=2 and Ξ΄=3 Tomimatsu-Sato solutions with universal normalization.

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We investigate trilinear structures as a natural extension of the Hirota bilinear formalism in integrable systems. While bilinear equations are associated with Grassmannian geometry and Pl\"ucker relations, trilinear equations suggest a higher algebraic structure involving three-slot couplings of tau functions. Focusing on the stationary axisymmetric Einstein equations, we show that when the Ernst potential is written in a tau-ratio form, the nonlinear equation decomposes into a cubic sector containing all second-derivative terms and a quartic gradient envelope. The cubic sector is identified with a YTSF-type trilinear kernel. We formulate a general trilinear kernel criterion and apply it to the Tomimatsu--Sato solutions. In particular, we demonstrate that the $\delta=3$ solution possesses the same trilinear kernel structure as the $\delta=2$ case, with a universal normalization up to a constant factor. These results suggest that the trilinear kernel represents a universal structure governing the highest-derivative sector of the Ernst system, providing a new perspective on integrability beyond the bilinear hierarchy.
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nlin.PS 2026-05-07

Distance-matrix spectra preserve static template during ring collapse

Frustrated Dynamics of Distance Matrices

Mass redistribution inside the fixed BBS shape flags the fast transition from uniform points to a one-dimensional ring on the sphere.

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We introduce the Frustrated Distance Matrix (FDM) model, a dynamic extension of the static distance-matrix ensemble on S^2 analyzed by Bogomolny, Bohigas, and Schmit (BBS). Its entries are pairwise geodesic distances between N Brownian particles on the sphere evolving under quenched random pairwise couplings linear in those distances. Where the static BBS theory recovers geometric information about the underlying manifold from spectra of distance matrices on i.i.d.\ samples, the time-resolved FDM spectrum carries information about structural changes of the underlying point process. The particle dynamics realize one such change: a fast collapse from a uniform configuration onto a one-dimensional ring, followed by slow rotational drift of the ring orientation; the particle-level picture provides the ground truth against which spectral diagnostics are calibrated. We find that the static BBS template is preserved at every time, with the dynamics entering as a redistribution of spectral mass within that template, sharp enough to flag ring formation. We propose self-averaging of the bulk density as the mechanism behind this preservation, verified by an i.i.d.-resample comparison, and extract a small set of spectral diagnostics of the structural change computable from the distance matrix alone. We suggest that our diagnostics can be applied in other similar inverse-problem settings: financial correlation matrices, graph and network adjacency spectra, similarity matrices in molecular dynamics, and dynamics on parameter manifolds.
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nlin.PS 2026-05-07

The paper shows that a single morphogen diffusing through layered two-dimensional media…

Patterns in Time and Space from a Single Morphogen via Nonlinear Layering

Nonlinear coupling across layers allows a single morphogen to produce Turing, Hopf, and Turing-wave instabilities in a reduced…

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Spatial and temporal pattern formation in reaction-diffusion systems is typically studied with two or more equations, as scalar reaction-diffusion equations confined to convex domains do not admit stable inhomogeneous states in time or space on long timescales. Here, we show that a single morphogen diffusing across layered two-dimensional media, with nonlinear coupling between layers, is able to generate stable patterns in time and space. This $N$-layer model is analysed via a thin-domain limit, which reduces to an $N$-component reaction-diffusion system on a homogeneous one-dimensional domain. This reduced model can be analysed via linear stability techniques, showing that non-diffusive, or reactive, coupling between regions is necessary for pattern-forming instabilities, at least in the reduced model. This reduced system can exhibit Turing, Hopf, and Turing-wave instabilities, with emergent structures that are numerically shown to persist even away from the thin-domain regime of the full 2D single-morphogen system. These results suggest that heterogeneous stratification and nonlinear coupling can broaden the class of systems which exhibit complex spatiotemporal behaviours, which may be relevant in scenarios where only a single morphogen is known to act.
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nlin.PS 2026-05-07

Stronger initial noise raises rogue wave odds in GI equation

Rogue wave statistics and integrable turbulence in the Gerdjikov-Ivanov equation

Simulations show disturbance intensity speeds chaos, shifts turbulence type, and fattens extreme-wave tails with lasting spectral asymmetry.

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This paper numerically investigates the statistical properties of rogue waves and their generation mechanisms in integrable turbulence, taking the Gerdjikov-Ivanov (GI) equation as the research object. The eigenvalue spectra of the analytical solutions and the chaotic wave field are calculated using the Fourier collocation method. Subsequently, taking a plane wave with random noise as the initial condition, the evolution of chaotic wave fields is simulated using the split-step Fourier (SSF) method. Numerical results show that the larger the initial disturbance intensity, the faster the wave field converges to a chaotic state, and the higher the peak amplitude after convergence, the higher the tail of the probability density function, and the significantly higher probability of rogue wave occurrence. Moreover, as the initial disturbance intensity increases, the turbulence type transitions from breather turbulence to soliton turbulence. In addition, the evolution of the wave-action spectrum is studied. The research has found that the wave-action spectrum of the GI equation shows an asymmetric distribution during the time evolution process, and this asymmetry persists even after the system reaches a steady state.
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nlin.AO 2026-05-07

Nonequilibrium efficiency peaks at phase transitions

Thermodynamic efficiency of self-organisation in nonequilibrium steady states

In active Ising models thermodynamic and inferential efficiencies diverge with distance from equilibrium.

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Active matter generates order or patterns through nonequilibrium dynamics. An open research challenge is to determine how efficiently a nonequilibrium self-organising system can convert consumed energy into macroscopic order. We study an information-theoretic quantity that directly addresses this challenge by estimating the entropy reduction induced by a small control-parameter perturbation, relative to the generalised work required for the perturbation. This quantity has previously been considered mainly in an equilibrium or near-equilibrium context, and here we extend this framework and apply it to two nonequilibrium self-organising systems: persistent and active Ising models. We observe that the thermodynamic efficiency of nonequilibrium systems maximises at phase transitions, as in equilibrium systems. Furthermore, we compare thermodynamic efficiency and inferential efficiency across control parameters. While these two quantities are equal in equilibrium as a consequence of the fluctuation-dissipation theorem, we report that they diverge out of equilibrium, and the gap reflects how far the system is from equilibrium.
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nlin.CD 2026-05-06

Transformers fail to predict dynamical system collapse in new regimes

Can Transformers predict system collapse in dynamical systems?

Trained only on stable parameters, they miss transitions that reservoir computing models detect.

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Transformer architectures have recently surged as promising solutions for nonlinear dynamical systems, proposed as foundation models capable of zero-shot dynamics reconstruction and forecasting. Despite this success, it remains unclear whether they can truly serve as reliable digital twins of dynamical systems, i.e., whether they capture the underlying physical dynamics in distinct parameter regimes, especially in parameter regimes from which no training data is taken. For parameter-space extrapolation in nonlinear dynamical systems, reservoir computing has demonstrated broad success, as proper training can turn it into an intrinsic dynamical system capable of capturing not only the dynamical climate of the target system but more importantly, how the climate changes with parameter. Transformers, in contrast, rely on permutation-invariant attention mechanisms that can limit their ability to capture how temporal structure changes with parameter. To determine if Transformers have the capability of dynamics extrapolation, we take predicting catastrophic collapse, which occurs when a bifurcation parameter crosses a critical threshold, as a benchmark task. Models are trained on trajectories in normal parameter regimes and then tested on parameters in an unseen regime with system collapse. Our results show that Transformers, across configurations, consistently fail to capture collapse, while reservoir computing reliably predicts the transitions. This surprising finding raises questions about the generalization ability of Transformers to dynamical systems, a topic warranting future research.
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nlin.CD 2026-05-06

Truncated memory in fractional oscillators creates effective delays

Gr\"unwald--Letnikov Memory Truncation in a Fractional Duffing Oscillator: Coherence Loss and Effective Delay Complexity

Trajectory comparisons reveal non-monotonic memory thresholds and the number of positive delays needed to match the kernel spectrum.

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We investigate the dynamical and analytical consequences of truncating the Gr\"unwald--Letnikov memory term in a fractional Duffing oscillator. The truncated memory is treated not merely as a computational approximation, but as a finite-memory modification of the underlying dynamical system. We define a coherence-loss time from direct comparisons between full-memory and truncated-memory trajectories, and use it to extract critical truncation thresholds in parameter planes involving the forcing amplitude and the fractional order. The results reveal strongly non-monotonic memory thresholds, showing that the retained memory required to preserve coherence depends on the forcing regime, the fractional order, and the nonlinear sensitivity of the dynamics. We also derive a local characteristic equation for the truncated GL kernel. A minimal one-delay approximation produces a formal negative delay, indicating that a single causal delay is structurally insufficient. This motivates a positive-delay exponential representation of the finite-memory kernel. The minimum number of positive-delay modes required to reach a prescribed spectral accuracy defines an operational delay-complexity measure, $r_{\min}$. Overall, the truncated GL kernel emerges as an intermediate object between distributed fractional memory and delay-type dynamics, with a local spectral structure that controls both coherence loss and effective delay complexity.
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nlin.CD 2026-05-06

Delayed feedback compensates for missing nonlinearities in linear optical reservoirs

Understanding Task Performance of Time-Multiplexed Optical Reservoir Computing via Polynomial Expansion

Transient coupling provides access to higher-order terms through multi-step integration, improving performance but requiring more virtualθŠ‚η‚Ή.

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We investigate the computational potential and limitations of a passive linear optical reservoir with a photodetector at the optical-to-electrical interface as the sole source of nonlinearity. In contrast to conventional nonlinear reservoirs, where transient dynamics and delay jointly enhance complexity and distribute nonlinear responses, the proposed linear architecture isolates these contributions, as intrinsic nonlinear spreading is absent. We thus provide a framework that enables the independent and systematic analysis of key factors, including nonlinear transformations, transient dynamics, and time-delay effects, as well as their interactions. By explicitly identifying the contributing monomials for different tasks, we establish the relationship between task requirements and the nonlinearity provided by the system. Incorporating transient coupling and delayed feedback is shown to significantly enhance performance and attractor reconstruction capabilities by compensating for missing higher-order nonlinearities through access to multi-step integration schemes. This improvement, however, comes at the cost of requiring a larger number of virtual nodes.
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nlin.SI 2026-05-05

Infinite conservation laws prove integrability of negative wave hierarchies

Negative Hierarchy of Hydrodynamic Type Equations

Explicit construction for shallow water waves and dispersionless Toda lattice negative hierarchies confirms their integrability.

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The negative integrable hierarchies of shallow water waves and dispersionless Toda lattice equations are considered. The integrability is shown by explicit construction of an infinite set of conservation laws.
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nlin.AO 2026-05-04 3 theorems

Slightly subcritical agents enable collective criticality

Emergent Macro-Criticality from Micro-Critical Agents

In light-signal multi-agent systems, micro regimes slightly below criticality support scale-free group avalanches over wider network ranges.

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Criticality has been proposed as a key principle underlying complex behavior in biological and artificial systems; however, how criticality translates from individual dynamics to collective behavior remains unclear. We study this question using a multi-agent system with spatially constrained interactions in which agents sense neighboring light signals through exteroceptors and act by switching their own light on or off, thereby forming a dynamical interaction network at the macroscopic level. The agents' internal states are themselves governed by a reservoir dynamical system at the microscopic level. By varying the microscopic parameters around dynamical criticality, together with the macroscopic interaction topology, we systematically investigate the relation between the two levels. We find that near-critical dynamics within individual agents is not sufficient to produce collective critical-like avalanche statistics. Instead, scale-free behavior depends on the effective connectivity of the macroscopic interaction network, which controls activity propagation. As a result, macroscopic critical-like dynamics are enabled by microscopic regimes that deviate from criticality, with the required deviation depending on the properties of the interaction network. Investigating this relation, we find that slightly subcritical micro-level regimes support near-critical dynamics across a wider range of macroscopic parameters. These results show that in this multi-agent system, collective near-critical behavior depends on the interplay between internal dynamics and the interaction structure that governs activity propagation.
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nlin.CD 2026-05-04

This paper claims that dynamical chaos arises as the spontaneous breaking of a…

The Supersymmetric Origin of Chaos and its Hidden Topological Order

Chaos is the spontaneous breaking of topological supersymmetry inherent to continuous-time dynamical systems, manifesting as hidden…

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Dynamical chaos is a term that encompasses a wide range of nonlinear phenomena such as turbulence, neuronal avalanches, weather patterns, and many others. However, despite much work in the field of chaos, its fundamental physical origin still remains not fully understood. In this perspective we report on recent studies showing that chaos is the realization of one of the most fundamental principles in physics: spontaneous symmetry breaking also known as spontaneous ordering. In the present context, the symmetry involved is a topological supersymmetry inherent to all continuous-time (stochastic) dynamical systems. Chaos is then truly a manifestation of order of topological origin potentially encoding a sort of long-range information hidden beneath its apparent unpredictability. We finally argue that this point of view may have far-reaching implications well beyond chaotic dynamics.
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nlin.CD 2026-05-04

Reservoir tuning via distribution matching recovers Lyapunov exponents

Optimizing Reservoir Computing for Reconstructing Ergodic Properties

Prediction-time tuning fails to ensure correct long-term statistics, but distribution matching succeeds.

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Reservoir computing is a powerful framework for modeling dynamical systems due to its universality and computational efficiency. However, a major challenge is achieving a forecast with accurate long-time statistics, or climate, which is essential for inferring ergodic properties such as Lyapunov exponents. A common approach is to optimize the reservoir's macroscopic parameters, such as the spectral radius, by maximizing prediction time. But here we show that even predictions accurate over multiple Lyapunov times do not guarantee the correct long-time statistics. Instead, we choose reservoir properties by minimizing the error in the reconstructed invariant distribution (or its projections), which is easily available from data. We demonstrate that this approach reproduces the Lyapunov exponents of model dynamical systems, including the logistic and standard maps, as well as the double pendulum, even with partial observations. We further show that recurrent connections, and resulting reservoir memory, are only required in the partially-observed case. We introduce a temporal scaling which reliably separates system and reservoir dynamics. In the posture time series of the nematode C. elegans we show that our approach quantitatively reproduces a chaotic behavioral attractor, but this requires a further constraint on the maximal conditional Lyapunov exponent to ensure the reservoir remains consistently synchronized to the complex biological input.
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nlin.AO 2026-05-04

Kuramoto model on higher-D torus has first-order sync transition

Kuramoto model on the D-dimensional torus

Incoherent state stays stable always while sync appears via saddle-node, unlike continuous 1D case

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We propose a generalization of the Kuramoto model of interacting oscillators in which the particles move on the surface of a $D$-dimensional torus. In contrast with the traditional one-dimensional version, this model has a first order phase transition. We establish its mean field dynamics by means of a multidimensional Ott-Antonsen ansatz, and show that synchronization arises from a saddle-node bifurcation, while the incoherent state is always stable. Our theoretical calculations are validated by numerical simulations.
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nlin.CD 2026-05-04

Circuit experiments capture spectral signatures of bifurcations

Experimental Acquisition and Verification of Spectral Signatures of Dynamic Bifurcations

Automated analog setups produce diagrams matching numerical predictions for period-doubling and quasiperiodicity despite noise.

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Spectral bifurcation diagrams (SBDs) have recently emerged as an efficient tool for identifying dynamical transitions in nonlinear systems through frequency-domain analysis. Previous studies have been limited to numerical investigations, and the experimental realization of SBDs has remained unexplored. In this work, we develop an automated framework using analog electronic circuits and data acquisition (DAQ) systems to obtain SBDs from real-time measurements. The method enables controlled parameter variation and simultaneous acquisition of time-series data for spectral analysis. Using this approach, we experimentally capture characteristic spectral signatures of dynamical bifurcations, such as period-doubling, quasiperiodicity (two- and three-frequency), and torus length-doubling. The experimental results show strong qualitative agreement with the numerical predictions, despite noise and parameter mismatches. This study establishes SBD as an effective tool for the experimental analysis of nonlinear dynamical systems.
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nlin.SI 2026-05-01

Schwarzian KP recast as bilinear equation on KP tau-function pairs

Bilinear formalism for Schwarzian KP and Harry Dym hierarchies

Linear combinations of the pair stay KP tau-functions, and this yields Harry Dym via Lax-Sato formulation.

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We consider the Schwarzian KP and Harry Dym hierarchies in the framework of the bilinear formalism which is well known for such integrable hierarchies as KP, modified KP, BKP, Toda lattice and other. We show that, similarly to the bilinear formulation of the modified KP hierarchy, the Schwarzian KP can be reformulated as an integral bilinear equation for a pair of KP tau-functions with the property that any linear combination of them is again a tau function of the KP hierarchy. The Harry Dym hierarchy is then obtained as the Lax-Sato formulation of the SchKP one. The close connection with Backlund-Darboux transformations for integrable hierarchies is also discussed. Besides, it is shown that the SchKP hierarchy has a natural embedding into the multi-component KP hierarchy.
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nlin.CD 2026-05-01

Elliptical addition reduces chaos threshold for oval billiards

Critical parameters of an oval billiard with an elliptical component

The derived formula shows how the elliptical strength shifts the onset of global chaos lower, while aligned phases can bring back regular

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We explore the critical parameters responsible for the transition from integrability to chaos in a family of billiards combining elliptical and oval deformations. Unlike standard oval billiards, where a known critical parameter governs the destruction of the last invariant curve, the introduction of an integrable elliptic component yields a second deformation axis. We derive an analytical expression for the critical parameter in this combined system and validate it numerically using Slater's theorem, showing that increasing the elliptical component lowers the critical threshold for global chaos. Moreover, we uncover a previously unexplored regime: when the two deformation components are in phase, the elliptic contribution progressively suppresses chaos, leading to the restoration of invariant curves and periodic orbits. A first-order analytical approximation confirms this behavior, supported by numerical validation. Our results reveal how the interplay between distinct boundary deformations enriches phase-space organization and offers enhanced controllability of chaotic dynamics in billiard systems.
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nlin.SI 2026-05-01

RH solver computes coupled mKdV solutions directly at any point

Numerical inverse scattering transform for the coupled modified Korteweg-de Vries equation

The 3x3 matrix problem is deformed in three regions using steepest descent to avoid time-stepping errors in long simulations.

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This paper develops the numerical inverse scattering transform (NIST) framework for the coupled modified Korteweg-de Vries (mKdV) equation based on its associated Riemann-Hilbert problem. The coupled system gives rise to a $3\times3$ matrix-valued Riemann-Hilbert problem, whose jump matrix and scattering data have a more involved structure than in the scalar case. This matrix setting makes the extension of NIST to the coupled system nontrivial, both in the direct scattering computation and in the numerical solution of the inverse problem. Within this framework, the scattering data are first computed by solving the matrix direct scattering problem using a Chebyshev collocation method with suitable mappings. The Deift-Zhou nonlinear steepest descent method is then used to analyze and deform the oscillatory Riemann-Hilbert problem. In particular, the phase function admits two stationary points symmetric about the origin, and the analysis leads to a division of the $(x,t)$-plane into three regions with corresponding contour deformations. Compared with traditional numerical methods, the NIST computes the solution directly at prescribed spatial and temporal points without relying on time-stepping. Numerical experiments illustrate the performance of the proposed NIST in long-time simulations and indicate that it captures the main asymptotic features of the coupled mKdV solutions.
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nlin.CD 2026-04-30

Astrocytes generalize Arnold tongues as star regions in parameter planes

Astrocytes: Arnol'd Tongues Generalization in Dynamical Systems' Parameter Plane

These branching, multi-vertex structures mark regular periodic zones amid chaos and show self-similar hierarchies in systems like the Zeeman

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We discovered generalized structures, named astrocytes due to their shape, that constitute a defined region characterizing regular behavior within the parameter plane (PP) of dynamical systems (DSs). Morphologically, they are characterized by a branch and a soma with several vertices (arms) and sometimes with multiple periodicities. A bunch of infinite astrocytes emerge through their branches from a region, in general, of low periodicity. Astrocytes are embedded in a quasiperiodic-chaotic scenario. The soma complexity (number of vertices) determines a kind of hierarchy of the astrocytes; moreover, bunches of subsequent structures from the astrocyte have been emphasized, revealing a self-similarity property. We conducted a detailed analysis in a Zeeman laser model, but we also observed astrocytes in many other DSs. The multiperiodicity exhibited by the astrocytes in their soma gives rise to harlequin dress-like patterns and tri-, quad-, and quint-critical points, which indicate the coexistence of different higher-order periodicities. In the concave borders of the soma, a doubling cascade of quint-points emerges as a bifurcation in the PP, defining regions of ordered sequences of higher periodicity in the route to chaos.
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nlin.SI 2026-04-30

Lax triples of KdV and NLS yield scalene Yang-Baxter maps

Scalene Yang-Baxter maps and Lax triples

The maps solve a generalized set-theoretic Yang-Baxter equation through matrix refactorization problems.

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We study a generalisation of the set-theoretic Yang-Baxter equation and investigate the connection between its solutions and matrix refactorisation problems. We refer to such solutions as scalene Yang-Baxter maps. Moreover, we construct scalene Yang-Baxter maps associated with integrable equations of KdV and NLS type.
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nlin.SI 2026-04-30

Gauge transforms simplify Lax pairs and build Miura maps for difference equations

On matrix Lax representations for (1+1)-dimensional evolutionary differential-difference equations

General theory for evolutionary differential-difference systems yields new two-component integrable equations linked by explicit discrete Mi

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Differential-difference matrix Lax representations (Lax pairs), gauge transformations, and discrete Miura-type transformations (MTs) belong to the main tools in the theory of (nonlinear) integrable differential-difference equations. For a given equation, two matrix Lax representations (MLRs) are said to be gauge equivalent if one of them can be obtained from the other by applying a matrix gauge transformation. Generalizing and extending several previous works on MLRs and MTs, we present new results on the following problems: - When and how can one simplify a given MLR by means of gauge transformations? - How can one use MLRs and gauge transformations for constructing MTs? - A MLR is called fake if it is gauge equivalent to a trivial MLR. How to determine whether a given MLR is not fake? We consider the general (1+1)-dimensional evolutionary differential-difference case when a MLR can depend on any shifts of dependent variables and can be non-autonomous. As applications and illustrations of the presented general theory, we construct several new two-component integrable equations (with new MLRs) connected by new MTs to known integrable equations from the papers [S. Konstantinou-Rizos, A.V. Mikhailov, P. Xenitidis, J. Math. Phys. 2015], [E. Mansfield, G. Mari Beffa, Jing Ping Wang, Found. Comput. Math. 2013]), including non-autonomous examples.
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nlin.SI 2026-04-30

Coupled rogue waves split into two distinct coexisting patterns

Coexistence of two distinct rogue wave patterns in the coupled nonlinear Schr\"odinger equation

High-order solutions develop separate regions each with a different fundamental wave type, shiftable by parameter choice

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This paper investigates the asymptotic behavior of high-order vector rogue wave (RW) solutions of the coupled nonlinear Schr\"odinger (CNLS) equation in the presence of multiple large internal parameters. We report several new high-order RW patterns in the CNLS system, including double-sector, double-heart, and mixed sector-heart configurations. The main novelty is that each RW pattern contains two distinct regions in which two different fundamental first-order RWs coexist simultaneously, potentially appearing as bright (eye-shaped) versus four-petaled or dark (anti-eye-shaped) forms. These two regions are respectively associated with the simple root structures of two different Adler--Moser polynomials: each region consists of well-separated first-order RWs in one-to-one correspondence with the simple roots of the associated polynomial. In addition, by tuning certain free parameters, the two regions of the RW pattern can be shifted to arbitrary locations in the $ (x,t) $-plane. This flexibility, together with the rich simple-root structures of Adler--Moser polynomials, enables the systematic generation of a much broader family of structured RW patterns in the CNLS equation.
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nlin.CD 2026-04-30

One machine reconstructs bifurcation diagrams for two chaotic systems

Inferring bifurcation diagrams of two distinct chaotic systems by a single machine

System-label and parameter-control channels let a reservoir computer generalize from partial data to full diagrams

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We propose a dual-channel reservoir-computing scheme for inferring the dynamics of two distinct chaotic systems with a single machine. By augmenting a standard reservoir with a system-label channel and a parameter-control channel, the machine can be trained from time series collected from a few sampled states of the two systems. We show that the trained machine not only predicts the short-time evolution of the sampled states, but also reproduces the long-term statistical properties of unseen states, thereby enabling reconstruction of the bifurcation diagrams of both systems from partial observations. The effectiveness of the scheme is demonstrated for the Lorenz and R\"ossler systems in numerical simulations and for the Chua and Rossler circuits in experiments. Functional-network analysis further shows that the two target systems are encoded by distinct dynamical patterns in the reservoir. These results extend multifunctional and parameter-aware reservoir computing, and provide a route to data-driven inference of multiple nonlinear systems using a single machine.
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nlin.PS 2026-04-30

Poles in nonlinear waves create monopoles of charge 3/2 and 5/2

Dirac monopole potentials with high charges underlying nonlinear waves

Simple poles and third-order poles of density functions produce higher quantized charges than zeros, shown via rogue waves and solitons.

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We investigate topological vector potentials underlying the phases of nonlinear waves by performing Dirac's magnetic monopole theory in an extended complex plane, taking into account self-steepening effects while ignoring the usual cubic nonlinearities. We uncover that the simple poles and third-order poles of the density function constitute virtual monopole fields with higher charges $\pm3/2$ and $\pm5/2$, respectively. These results are in sharp contrast to the previous findings, where the simple zeros of the density function yield charges $\pm1/2$. We choose scalar and vector rogue waves as well as bright solitons to demonstrate the Dirac monopole potentials. These results confirm a series of quantized magnetic charges for virtual monopoles underlying nonlinear waves, and reveal new relations between poles of density functions and topological charges.
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nlin.PS 2026-04-30

Turing patterns on fixed lattices respond to mechanical stress

Turing patterns on non-fluctuating surfaces under mechanical stresses

Finsler modeling with directional internal vectors shows patterns change under force the same way they do on fluctuating membranes, and lets

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This paper presents a numerical investigation of Turing patterns (TPs) utilizing the reaction-diffusion equation for the activator $u$ and the inhibitor $v$ on two- and three-dimensional lattices, discarding vertex fluctuations. The absence of vertex fluctuations means the absence of positional movement of $u$ and $v$. Consequently, $u$ and $v$ have values at spatially discrete points, such as the pigment cells in zebrafish and sea shells. Furthermore, the mechanical property is implemented through the Finsler geometry modeling technique. This technique incorporates the internal degree of freedom $\vec{\tau}$, corresponding to the direction of mechanical stress. Additionally, a stress formula based on Gaussian bond potential is shown to be well-defined on the non-fluctuating lattices, and therefore, it enables the calculation of entropy for capturing the stress relaxation phenomenon in a manner analogous to that on fluctuating surfaces. The results of the study indicate that these biological patterns also exhibit responses to external mechanical forces similar to TPs on fluctuating membranes.
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nlin.PS 2026-04-29

Space-time pulse splits into soliton train in multimode fiber

Space-time excitation creates soliton trains in multimode fibers

One input with topological charge β„“ produces |β„“| + 1 multimode solitons whose energies and modes are set by β„“.

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In this work, we show that injecting a single space-time-coupled light pulse-beam into a multimode graded-index fiber generates a train of multimode solitons. Space-time couplings excite the spatial modes with distinct temporal profiles. Due to nonlinear interactions, with a properly chosen input power these profiles split into several unique multimode solitons. In the case of a spatially chirped input pulse, two solitons composed of modes $LP_{01}$ and $LP_{11}$ are formed. In the case of the injection of a space-time optical vortex, characterized by its topological charge $\ell$, a train composed of $|\ell| + 1$ multimode solitons is generated. Their energy and modal composition are directly determined by the absolute value of the topological charge.
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nlin.CD 2026-04-29

Negative average power sustains resonance via fractional memory

Transmitted and Storage-Dominated Resonance in Fractionally Damped Unidirectionally Coupled Duffing Oscillators

In unidirectionally coupled Duffing oscillators, fractional damping lets the receiver oscillate strongly while time-averaged coupling power,

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This paper investigates resonance transmission in two unidirectionally coupled Duffing oscillators with fractional damping, where the driver is harmonically forced and the receiver is connected through a linear coupling spring. Particular attention is paid to how fractional damping in the receiver modifies amplitude amplification, energy redistribution, and the structure of the coupled response. The numerical results reveal a clear distinction between transmitted resonance, associated with a coupling-power balance consistent with direct energy transfer through the coupling spring, and storage-dominated resonance, in which the receiver still exhibits a pronounced oscillatory response while the time-averaged coupling power becomes negative under the adopted convention. In this latter regime, fractional memory promotes temporary energy accumulation within the receiver--coupling subsystem, followed by partial release through the coupling spring without any feedback on the driver dynamics. We further show that detuning the receiver natural frequency enhances the interaction between the lower-frequency transmitted response and the higher-frequency coupled response, leading to a superposed resonance regime with increased receiver amplitude, stronger localization, and sharper response. The roles of the fractional order, coupling strength, and receiver natural frequency are systematically analyzed through frequency-response curves and parametric maps. Overall, the results show how fractional memory can be used to tune resonance transmission, energy localization, and amplified response in coupled nonlinear oscillators.
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nlin.CD 2026-04-28

LAVD maxima plus contraction tests locate rotating contracting structures

Lagrangian Rotating Contracting Structures

The pairing works in strongly deforming flows where level-set geometry no longer marks vortical regions

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We identify materially defined regions in unsteady two-dimensional flows that combine finite-time contraction with elevated accumulated intrinsic rotation along trajectories, which we term \emph{Lagrangian rotating contracting structures} (LRCS). These regions are detected using existing objective diagnostics -- the Lagrangian-averaged vorticity deviation (LAVD) together with direct tests of material contraction -- without relying on the geometry of LAVD level sets. In strongly deforming flows, LAVD maxima need not correspond to vortical regions or be enclosed by regular level sets, rendering geometry-based identification unreliable. Nevertheless, regions exhibiting inward spiraling motion and contraction can be extracted by combining LAVD with a contraction criterion. Applications to atmospheric and oceanic flows show that such behavior arises both in twisted LAVD fields generated at submesoscales and in mesoscale flows where it is enhanced by inertial effects, with finite-time contraction providing the dynamical constraint that isolates materially organized regions with elevated intrinsic rotation.
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nlin.CD 2026-04-28

Regression estimates resilience in non-stationary systems

Estimating the Resilience of Non-Stationary Systems

Handles seasonal forcing, data gaps, and irregular sampling as a direct replacement for autocorrelation-based estimates.

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A wide body of work has applied the concept of critical slowing down to estimate the stability of different Earth system components. Most of them -- such as global vegetation -- are inherently non-stationary, for example due to strong seasonal forcing, which complicates the estimation of their resilience to external perturbations. Here, we introduce a new method to account for non-stationarity in estimating resilience for diverse synthetic and real-world data sets via a regression-based formulation of the Langevin Equation. Our method does not require extensive data pre-processing, is robust to gaps in the data record, and does not require regular time sampling. We further show that our method can incorporate time-varying data uncertainties, recover uncertainty bounds in stability estimates, and can be natively extended to examine spatial systems. Our method is a drop-in replacement for widely-used autocorrelation-based resilience estimates, and can be widely applied across Earth system components.
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nlin.PS 2026-04-28

Repulsive interactions stabilize dark-bright solitons in 2D BEC mixtures

Impurity localization, and collision properties of symbiotic dark-bright solitons in superfluid-impurity system

Phase differences between bright components decide whether colliding solitons merge or repel.

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We investigate the dynamics of a binary mixture of Bose-Einstein condensates in the impurity limit -- where one component is dilute enough to be treated like an impurity -- and confined to two dimensions. Using the mean-field coupled Gross-Pitaevskii equations, we find that the binary mixture supports the formation of stable symbiotic dark-bright solitons when the inter- and intra-component interactions are repulsive. We further study the interaction between solitons and observe that the solitons undergo merging and repulsion depending on the relative phase between the bright component of the composite structure.
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nlin.SI 2026-04-28

Hirota-Miwa equations collapse to seven distinct models

Discrete integrable equations with three independent variables

Darboux reductions link Toda, semi-discrete and fully discrete classes while preserving integrals, producing Lax pairs for each.

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In this paper, we study nonlinear integrable equations with three independent variables of the following types: Toda-type lattices, semi-discrete lattices, and fully discrete Hirota-Miwa type models. It is shown that integrable equations of all three types admit reductions in the form of Darboux-integrable hyperbolic systems. It is important that the transition from one class to another is carried out by means of discretization (continualization) of the above-mentioned reductions with preservation of characteristic integrals. In other words, at the level of reductions, one can establish some correspondence between the classes of 3D models under consideration. In the context of this correspondence, the authors managed to conduct a comparative analysis of the well-known list of integrable Hirota-Miwa type equations, containing 13 equations. It was established that some equations from this list are related by point changes of variables. As a result, the final list of known integrable Hirota-Miwa type equations was reduced to seven. One equation was obtained by discretizing the list of semi-discrete Toda-type equations using characteristic integrals in this paper, probably it is new. For all seven models, associated linear systems (Lax pairs) are given.
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nlin.AO 2026-04-27

Bayesian method recovers phase equations from traveling pattern data

Data-driven reconstruction of spatiotemporal phase dynamics for traveling and oscillating patterns via Bayesian inference

Time-series data yield the deterministic rules for pattern position and oscillation when noise is weak and symmetry holds.

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Building on the phase reduction theory formulated for reaction-diffusion systems with spatial translational symmetry, we develop a data-driven method that reconstructs the spatiotemporal phase dynamics of traveling and oscillating patterns. Spatiotemporal phase dynamics are described by spatial and temporal phases that represent the position and oscillation of the pattern, respectively. Using Bayesian inference, our method directly reconstructs phase equations from time-series data. When tested on simulation data from coupled Gray-Scott models exhibiting traveling breathers, the method accurately reconstructs the deterministic part of the phase equations in the weak-noise regime, in which the phase dynamics converge to a linearly stable fixed point.
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nlin.CD 2026-04-27

FTLE distributions match original chaos inside trained reservoirs

Finite-time Lyaponov analysis of a trained reservoir computer

High-dimensional maps reproduce intermittency and crisis signatures of the logistic map even when periodic orbits cannot be tracked directly

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We use finite-time Lyapunov exponent (FTLE) distributions to probe transition mechanisms in high-dimensional reservoir maps trained on low-dimensional chaotic dynamics across multiple regimes. While trained reservoirs accurately predict critical transitions and regime shifts, conventional analyses based on time series or bifurcation structure provide limited mechanistic insight, since distinct pathways in high dimensions can yield similar outputs. We show that FTLE statistics overcome this limitation. This is particularly important for interior crises, where direct identification of unstable periodic orbit collisions in the reservoir space is infeasible. Using the logistic map as a canonical example exhibiting intermittency, fully developed chaos, and crisis-induced transitions, we demonstrate that although such distinct regimes are difficult to characterize within the high dimensional reservoir space, their FTLE distributions are faithfully reproduced. This establishes FTLE analysis as a systematic and reliable framework for uncovering transition mechanisms in learned reservoir dynamics.
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nlin.SI 2026-04-27

Umemura roots predict rogue wave patterns

Rogue-wave and lump patterns associated with the third Painlev\'{e} equation

Large parameters in nonlinear SchrΓΆdinger and Boussinesq equations align rogue waves with polynomial roots from the third PainlevΓ© equation,

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We report rogue-wave and lump patterns associated with Umemura polynomials, which arise in rational solutions of the third Painlev\'{e} equation. We first show that in many integrable equations such as the nonlinear Schr\"odinger equation and the Boussinesq equation, when internal parameters of their rogue wave solutions are large and of certain form, then their rogue patterns in the spatial-temporal plane can be asymptotically predicted by root distributions of Umemura polynomials (or equivalently, pole distributions of rational solutions to the third Painlev\'{e} equation). Specifically, every simple root of the Umemura polynomial would induce a fundamental rogue wave whose spatial-temporal location is linearly related to that simple root, while a multiple root of the Umemura polynomial would induce a non-fundamental rogue wave in the $O(1)$ neighborhood of the spatial-temporal origin. Next, we show that in a certain class of higher-order lump solutions of the Kadomtsev-Petviashvili-I (KPI) equation, when their internal parameters are large and of certain form, then their lump patterns at $O(1)$ time can also be predicted asymptotically by root distributions of Umemura polynomials, where simple and multiple roots of the polynomial would give rise to fundamental and non-fundamental lumps in the spatial plane, respectively. These results reveal the importance of the third Painlev\'{e} equation in studies of nonlinear wave patterns. We also report a new transformation which turns bilinear rogue-wave solutions of the nonlinear Schr\"odinger equation to higher-order lump solutions of the KPI equation.
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nlin.AO 2026-04-27

Hydrodynamics make uniform micro-collectives look heterogeneous

Hydrodynamic interactions mask the true heterogeneity of a microscopic collective

Measured speeds register diversity even when all agents share identical motilities if fluid interactions dominate.

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Coordinated movement and self-organisation of active self-driven agents is common in nature and is seen across different scales, from herds of animals to collective motion in bacteria. Often, these systems are heterogeneous in composition, with different agents having different intrinsic motilities. Inferring these intrinsic characteristics and quantifying the level of heterogeneity in a collective system is crucial to understanding the observed emergent phenomena. However, when interaction effects dominate, i.e. the observed movement of an agent is strongly influenced by its interacting neighbours, inferring the intrinsic characteristics of agents becomes a challenge. We consider a collective system of agents that undergo purely physical interactions like collisions and long-range hydrodynamic interactions, which resembles a system of microswimmers immersed in a fluid medium. We incorporate heterogeneity into the system through variations in agent motility and examine how the perceived heterogeneity, inferred from measured speeds, depends on the strength of hydrodynamic interactions and the true intrinsic variability. The interplay between short-range collisions, long-range hydrodynamic interactions, and intrinsic heterogeneity makes the inference problem non-trivial. When hydrodynamic effects dominate, true heterogeneity is effectively masked, making even a homogeneous collective appear heterogeneous. The competing effects of collisions, which slow agents down, and hydrodynamic interactions, which enhance their motion, further complicate reliable inference. Hydrodynamic interactions also modify collision angles, rendering them more isotropic. Overall, the findings show highlight experimentally measured properties of microscopic collectives may not accurately reflect their true characteristics.
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nlin.PS 2026-04-27

Shock waves in spherical and cylindrical KdV-B follow stable superposition

Shock waves of spherical/cylindrical KdV-B: Asymptotic, stability, superposition

One-parameter families of diverging shocks have explicit asymptotics, are stable by conservation, and combine simply even if discontinuous.

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Spherical and cylindrical KdV-B equations have few known exact solutions, yet these solutions are hard to be interpreted physically. But these equations do have a family of diverging shock waves. Their properties such as asymptotic modes, stability, rules of their interactions/superposition are the subject of this paper. It gives a detailed asymptotic description of the one-parameter families of shock wave solutions and proves their stability using a conservation law. Based on these results, effective rules of superposition are obtained. Moreover these rules are applicable to a wide class of shock waves, in particular discontinuous. Typical examples are illustrated by graphs.
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nlin.AO 2026-04-27

Forcing eliminates standing waves in bimodal inertial oscillators

Interplay of inertia and external forcing in Kuramoto model

External drive competes with frequency bimodality to remove intermediate states and make the backward transition discontinuous.

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The impact of external forcing is well studied in the Kuramoto model without inertia, but remains unclear for inertial Kuramoto oscillators (KMI) with bimodal intrinsic frequency distributions. This article fills that gap, showing that competition between external forcing and intrinsic bimodality can suppress the intermediate standing wave states of bimodal KMI by entraining oscillators to the external forcing. Using a self-consistent analytical framework, we show that, for a bimodal distribution, forcing makes the backward transition discontinuous, unlike the continuous transition in the unimodal case. Further, for a bi-delta distribution, we derive a closed form expression for the backward solution branch. These results clarify how intrinsic frequency structure shapes the effect of external forcing, with implications for biological systems (e.g., photoreceptor and pacemaker cells) and for pinning-control strategies in multi-agent networks.
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nlin.PS 2026-04-24

Delays induce waves and breathing bumps in theta neuron rings

Dynamic solutions of next generation neural field models with delays

Hopf bifurcations in delayed continuum models create dynamic solutions whose parameter variation maps delay effects on neural patterns.

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We study networks of theta neurons arranged on a ring with delayed interactions. In the continuum limit the systems are described by next generation neural field models with delays. We consider distributed delays with both finite and infinite support, and conduction delays. The stability of spatially uniform and localized bump states is determined, and we find that they undergo Hopf bifurcations as parameters related to the delays are varied. These bifurcations create traveling waves and ``breathing'' bump solutions. These dynamic solutions satisfy self-consistency equations and we show how to efficiently solve these equations. Following traveling waves and periodic solutions as parameters are varied provides a global picture of the influence of different delays on pattern formation processes in spatially extended networks of theta neurons.
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nlin.SI 2026-04-23

Invariants test if spectral parameters in Lax pairs are essential

The gauge action on semi-discrete Lax representations and its invariants

Nontrivial lambda-dependence in any invariant means no gauge transformation can remove the parameter.

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Semi-discrete (differential-difference) matrix Lax representations (Lax pairs) play an essential role in the theory of integrable differential-difference equations. Fix a (1+1)-dimensional evolutionary differential-difference (semi-discrete) equation and consider matrix Lax representations (MLRs) of this equation. Two MLRs are said to be gauge equivalent if one of them can be obtained from the other by applying a (local) matrix gauge transformation. Gauge transformations (GTs) form an infinite-dimensional group, which acts on the set of MLRs of a given equation. Two MLRs are gauge equivalent iff they belong to the same orbit of this action. When one tries to establish integrability (in the sense of soliton theory) for a given equation, one is interested in MLRs which depend on a parameter (usually called the spectral parameter) such that the parameter cannot be removed by any GT. We introduce and study explicit invariants with respect to the action of GTs on the set of MLRs for a given (1+1)-dimensional evolutionary differential-difference equation with any number of components. Using these invariants, we obtain the following results: - Consider a MLR with a parameter $\lambda$. If at least one of the invariants computed for this MLR depends nontrivially on $\lambda$, then the parameter cannot be removed by any GT. - When we have two different MLRs for a given equation, we present necessary conditions for these two MLRs to be gauge equivalent. Our results on semi-discrete MLRs of differential-difference equations are inspired by results of S$.$Yu. Sakovich and M. Marvan on (continuous) zero-curvature representations of partial differential equations. A comparison with some of the results of S$.$Yu. Sakovich and M. Marvan is presented.
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nlin.CD 2026-04-23

Force field in MLC circuit triggers extreme events via attractor expansion

Extreme events in MLC circuit

Large chaotic attractor expansion follows period-merging intermittency and is confirmed by manifold and statistical analyses.

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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.
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nlin.SI 2026-04-22

New duality maps Hamiltonian to Lagrangian forms for integrable systems

Duality of Hamiltonian and Lagrangian formulations for integrable systems

Hamiltonian potential variables generalize the KdV trick and supply Lagrangian multiforms for gas dynamics and astigmatism equations.

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We introduce the concept of Hamiltonian potential variables to map Hamiltonian operators into symplectic operators in a dual space. This generalises the classical trick of switching to a potential variable to obtain a Lagrangian density for the Korteweg-de Vries (KdV) equation. Building on this concept, we present the Lagrangian structure for bi-Hamiltonian systems, discuss the Lenard scheme in the symplectic formalisms, and apply this to construct pairs of Lagrangian multiforms. We discuss the key model of the KdV equation and some dispersionless limits of it. We present a pair of Lagrangian multiforms for these equations, one of which is new. We also consider the examples of polytropic gas dynamics and the constant astigmatism equation, for which no Lagrangian multiforms were previously known.
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nlin.SI 2026-04-22

Orlov-Schulman symmetries extend self-dual conformal hierarchy

Orlov-Schulman symmetries of the self-dual conformal structure equations

They commute with the basic Lax-Sato flows, include Galilean and scaling cases, and arise from Riemann-Hilbert dressing.

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We construct Orlov-Schulman symmetries for the self-dual conformal structure (SDCS) hierarchy. We provide an explicit proof of compatibility of additional symmetries with the basic Lax-Sato flows of the hierarchy, and consider several simple examples, including Galilean transformations and scalings. We also present a picture of the Orlov-Schulman symmetries in terms of a dressing scheme based on the Riemann-Hilbert problem.
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nlin.PS 2026-04-22

Exact solitary waves in PT-symmetric Dirac equation independent of k

Generalized PT-symmetric nonlinear Dirac equation: exact solitary waves solutions, stability and conservation laws

PT-transition set by existence condition alone with conserved energy but restricted stability domain

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We derive an exact solitary wave solution for the $\PTb$-symmetric nonlinear Dirac equation with a scalar-scalar interaction. We consider a power-law nonlinearity of the form $|\bar{\Psi}\,\Psi|^{k}\,\Psi$ for positive values of $k$. The system's energy is conserved despite the presence of a gain-loss term, which is quantified by the parameter $\Lambda$. We show that the $\PTb$-transition point is defined by the solution's existence condition and is independent of the nonlinearity exponent $k$. Furthermore, momentum is conserved, although neither the canonical momentum nor the charge is a conserved quantity. A notable result is that the stationary solution, obtained from the continuity equations, exhibits nonzero momentum in its rest frame. We also derive a moving soliton solution, where the gain-loss parameter allows the soliton's velocity to be precisely chosen so that the moving soliton possesses zero momentum. Finally, we establish that the presence of a gain-loss mechanism and higher-order nonlinearity restrict the stability domain of the solutions.
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nlin.SI 2026-04-22

Explicit canonical coordinates separate Kaup-Kupershmidt HΓ©non-Heiles dynamics

Canonical separating coordinates in the generalized cubic H\'enon-Heiles systems

Bi-Hamiltonian geometry supplies both the coordinates and their conjugate momenta, splitting the four-dimensional system into two decoupled

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We study the three classical integrable generalized cubic H\'enon--Heiles systems -- Kaup--Kupershmidt, KdV$_5$, and Sawada--Kotera -- from the viewpoint of bi-Hamiltonian geometry and separation of variables. On the standard symplectic manifold $T^*\mathbb R^2$, we construct compatible Poisson deformations $P_1=L_XP_0$, compute the associated recursion operators $N=P_1P_0^{-1}$, and analyze the action of $N^*$ on the codistribution generated by the first integrals. This yields the corresponding control matrices, whose eigenvalues provide the separating coordinates. For the generalized Kaup--Kupershmidt case we carry out the construction explicitly: we determine a deformation vector field, the compatible Poisson tensor, the torsionless recursion operator, the control matrix, the separating coordinates, and, crucially, the conjugate momenta. We then derive the separated relations and write the Hamilton equations in canonical separated variables, thus decomposing the original Hamiltonian system into two separated subsystems. To the best of our knowledge, this explicit derivation of the separating variables and, in particular, of the conjugate momenta for the generalized Kaup--Kupershmidt system is new. For the KdV$_5$ and Sawada--Kotera cases we show how the same bi-Hamiltonian scheme applies, emphasizing both the common geometric mechanism and the features peculiar to each system. In this way, the three generalized cubic H\'enon--Heiles systems are treated within a unified framework based on compatible Poisson structures, recursion operators, control matrices, and Darboux--Nijenhuis coordinates.
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nlin.PS 2026-04-21

Nonlinear model supports traveling solitary waves on topological edges

Chiral solitary waves in a nonlinear topological insulator model

Chiral edge states persist and collide inelastically in this nonlinear topological model.

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An outstanding challenge in the field of topological insulators is the realization of nonlinear systems that support coherent traveling waves. Highly nonlinear lattices can suffer from significant radiation losses due to Peierls-Nabarro effects. In this work a nonlinear tight-binding model that supports robust traveling edge states is proposed and examined. This system possess a nontrivial local Chern topology and soliton-like states. When a traveling solitary wave collides with a stationary mode, the two are observed to interact inelastically. These results suggest future directions for the modeling, realization, and application of nonlinear Chern insulators.
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nlin.CG 2026-04-21

Game of Life patterns show temporal retention as biosignature

Temporal Retention of Information as a Biosignature

Persistent evolutions suggest information retention over time as a marker for life in abstract systems.

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Previous publications by the authors put forward the argument that Lifelike Cellular Automata can be treated as a bona fide example of livingness in and of themselves, not simply a toy analogue to biological life. Traits known to be indicative of biological life, biosignatures, were identified in informational form as particular outlier traits of the ruleset for the lifelike cellular automata known as Conways Game of Life. This publication reverses that logic, looking at a known outlier trait of Conways Game of Life, its very long-lasting evolutions, and using this to point towards temporal retention as an informational biosignature concept.
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nlin.AO 2026-04-21

Unified edge weights give tighter stability conditions for networks

A Unified Theory of Edge Weights: Stability of General Laplacian Networks from Matrix Phases and Asymmetry Rayleigh Ratios

A general formulation for directed and multidimensional edges produces less conservative bounds in power grids and oscillator models.

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We study the properties and stability of networks with arbitrary Laplacian coupling. Classic approaches to studying networked systems require unrealistic assumptions, including homogeneous node dynamics, one-dimensional and undirected edges, or constant edge weights. We develop a unified formulation of Laplacian-style couplings that drops these assumptions, providing a unified notion for the edge weights of adaptive, directed, and multi-dimensional edges. We show that the recently developed theory of matrix phases can capture essential stability properties of the network and its edges. We quantify the impact of the asymmetry of the higher-dimensional edge dynamics on the system's phase properties by introducing the Asymmetry Rayleigh Ratio. These theoretical advances allow us to derive new sufficient stability conditions for AC power grids, directed diffusion, and the Kuramoto-Sakaguchi model. The resulting conditions are less conservative than the specific results known for these systems.
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nlin.CD 2026-04-20

Analog circuit maps full bifurcation landscape of Duffing-Holmes

From order to chaos: Bifurcations and parameter space organization in an analog Duffing-Holmes circuit

High-resolution experiments trace period-doubling to chaos and multistable regions matching the ideal model.

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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, Poincar\'e 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.
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nlin.CD 2026-04-20

Coupled chaotic engines reduce to single engines with defined efficiency

The thermodynamic efficiency of coupled chaotic dissipative structures

Association laws for series and parallel couplings compute overall thermodynamic performance from the components

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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.
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nlin.AO 2026-04-20

Adaptive agents self-organize to ergodicity edge

Self-Organization to the Edge of Ergodicity Breaking in a Complex Adaptive System

At this boundary the system produces scale-free avalanches and higher collective rewards than any fixed regime.

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Self-organized criticality (SOC) is widely proposed as a fundamental mechanism for collective behavior, yet its role in objective-driven, heterogeneous adaptive systems underpinning real complex systems remains less understood. We introduce EvoSK, a minimal evolutionary model in which agents perform memory dependent reinforcement learning on a rugged Sherrington-Kirkpatrick landscape while the population evolves through extremal replacement of the least fit agents. We demonstrate that this coupled dynamics drives the system to a critical state residing on the transition boundary between ergodic and non-ergodic phases. At this boundary, the system exhibits scale-free evolutionary avalanches with a mean-field exponent $\tau \approx -1.5$, while simultaneously achieving collective rewards that surpass those of any manually finetuned, non-evolutionary regime. Our results provide a mechanistic link between the statistical physics of ergodicity breaking and the functional optimality of complex adaptive systems, suggesting that the edge of ergodicity breaking acts as a robust attractor for systems adapting on rugged, high-dimensional landscapes.
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nlin.PS 2026-04-20

Dark solitons keep intensity dips intact in nonlinear SSH lattices

Dark solitons in nonlinear Su-Schrieffer-Heeger lattices

The dips stay preserved in bulk and edges within various gaps, stable only when intracell coupling is dominant.

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The introduction of nonlinearities into lattices with topological band structures has led to the discovery of various types of solitons. The Su-Schrieffer-Heeger (SSH) lattice, as the most fundamental topological model, has been extended into the nonlinear regime. In particular, nonlinear edge states and bulk solitons exhibiting intensity humps against a zero background have been extensively studied in nonlinear SSH lattices. In this paper, we systematically investigate dark solitons in nonlinear SSH lattices. These dark solitons maintain a nonzero and constant background, featuring intensity dips either in the bulk of the lattice or at its edges, and residing spectrally in the semi-infinite gap or the middle finite gap. Regardless of the specific type of dark soliton, the intensity dip remains wellpreserved and is not affected by the band structure of the original linear lattice. Although the dark solitons we have identified are generally dynamically unstable across a broad range of parameters, several types exhibit linear stability when the intracell coupling is much larger than the intercell coupling. Our findings may provide valuable insights for the exploration of novel types of solitons in nonlinear topological lattices.
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nlin.CG 2026-04-20

Count data alone leaves diffusion parameters unidentified

When do trajectories matter? Identifiability analysis for stochastic transport phenomena

Trajectory records resolve structural non-identifiability in lattice random walk models of population movement.

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Stochastic models of diffusion are routinely used to study dispersal of populations, including populations of animals, plants, seeds and cells. Advances in imaging and field measurement technologies mean that data are often collected across a range of scales, including count data collected across a series of fixed sampling regions to characterize population-level dispersal, as well as individual trajectory data to examine at the motion of individuals within a diffusive population. In this work we consider a lattice-based random walk model and examine the extent to which model parameters can be determined by collecting count data and/or trajectory data. Our analysis combines agent-based stochastic simulations, mean-field partial differential equation approximations, likelihood-based estimation, identifiability analysis, and model-based prediction. These combined tools reveal that working with count data alone can sometimes lead to challenges involving structural non-identifiability that can be alleviated by collecting trajectory data. Furthermore, these tools allow us to explore how different experimental designs impact inferential precision by comparing how different trajectory data collection protocols affects practical identifiability. Open source implementations of all algorithms used in this work are available on GitHub.
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nlin.CG 2026-04-17

Game of Life patches become programmable semigroups

Measuring the Computational Power of Finite Patches of Cellular Automata

Mapping finite cellular automaton regions to transformation semigroups measures their interactive computational power and enables algebraic,

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Computational power can be measured by assigning an algebraic structure to a computational device. Here, we convert a small patch of Conway's Game of Life into a transformation semigroup. The conversion captures not only time evolution but also interactive operations. In this way, the cellular automaton becomes directly programmable. Once this measurement is made, we apply hierarchical decompositions to the resulting algebraic object as a way of understanding it. These decompositions are based on a macro/micro-state division inspired by statistical mechanics. However, cellular automata have a large number of global states. Therefore, we focus on partitioning the state space and creating morphic images approximations that can serve as macro-level descriptions. The methods developed here are not limited to cellular automata; they apply more generally to discrete dynamical systems.
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nlin.AO 2026-04-17

OPUC theory reduces multiharmonic oscillator populations

Low-Dimensional Reduction Theory for Populations of Globally Coupled Phase Oscillators with Multiharmonic Coupling: A Method Based on OPUC Theory

Orthogonal polynomials supply a framework that lowers the dimension of phase-oscillator models with arbitrary multi-harmonic couplings.

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Low-dimensional reduction theories, such as the Ott-Antonsen ansatz, have played a crucial role in the study of populations of globally coupled phase oscillators. However, most of these theories are applicable only to models in which the interaction is described by a single harmonic component, limiting their application to more realistic oscillator models. In this paper, by employing the theory of orthogonal polynomials on the unit circle (OPUC), we construct a framework that enables low-dimensional reduction for populations of globally coupled phase oscillators with multiharmonic coupling.
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nlin.CD 2026-04-16

MA integrals estimate secondary resonance widths without normalization

Melnikov-Arnold integrals and optimal normal forms

In the standard map, Melnikov-Arnold integrals taken from the original system give resonance sizes up to optimal order.

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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.
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nlin.SI 2026-04-16

Same PainlevΓ© surface from different weights but distinct dynamics

On the discrete Painlev\'e equivalence problem, non-conjugate translations and nodal curves

Non-conjugate elements in the Weyl group and nodal curves make some examples inequivalent despite shared D_5^{(1)} type.

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We consider several examples of nonautonomous systems of difference equations coming from semi-classical orthogonal polynomials via recurrence coefficients and ladder operators, with respect to various generalisations of Laguerre and Meixner weights. We identify these as discrete Painlev\'e equations and establish their types in the Sakai classification scheme in terms of the associated rational surfaces. In particular, we find examples which come from different weights and share a common surface type $D_5^{(1)}$ but are inequivalent in two ways. First, their dynamics are generated by non-conjugate elements of $\widehat{W}(A_3^{(1)})$. Second, some of the examples have associated surfaces being non-generic in the sense of having nodal curves. The symmetries of these examples form subgroups of the generic symmetry group, which we compute. In particular, we find $(W(A_1^{(1)})\times W(A_1^{(1)}))\rtimes \mathbb{Z}/2\mathbb{Z}$. These examples give further weight to the argument that any correspondence between different weights and the Sakai classification should make use of the refined version of the discrete Painlev\'e equivalence problem, which takes into account not just surface type, but also the group elements generating the dynamics as well as parameter constraints, e.g. those corresponding to nodal curves.
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nlin.CD 2026-04-15

Biomimetic scales produce chaotic bending vibrations at low amplitudes

Chaotic Flexural Vibrations in Biomimetic Scale Substrates

Contact jamming and textural asymmetry shift the route to chaos in scale-covered beams, allowing geometric tuning of the dynamics.

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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.
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nlin.CD 2026-04-15

Neutrino billiards model relativistic quantum chaos

Relativistic Quantum Chaos in Neutrino Billiards

Spin-1/2 particles confined in planar domains show how relativity changes chaos signatures, with graphene as a possible lab platform.

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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.
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nlin.CD 2026-04-15

Chaos enhances tunneling only via resonance assistance

Chaos and Quantum Tunneling

Review clarifies the regimes where chaotic regions increase quantum penetration of phase space barriers and traces the effect to specific me

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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.
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nlin.CD 2026-04-15

Lyapunov vector cascade yields perfect predictors for extreme events

Precursors of extreme events and critical transitions

Fast-slow nonlinear systems show a repeatable three-regime sequence before large excursions, giving two indicators with full precision and 0

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We propose a theory based on dynamical systems to explain and predict the occurrence of extreme events, of which critical transitions form a subset. In fast-slow nonlinear systems, we identify a cascade of events preceding extreme events: (i) a slow regime, in which the fast covariant Lyapunov vectors (CLVs) are both tangent to the fast eigenvectors and remain transversal to the slow subspace; (ii) a transition regime, in which the fast eigenvalues become neutrally stable while the fast CLVs are no longer tangent to the fast eigenvectors; and (iii) a critical regime, in which a strong spectral gap in the eigenvalues causes both fast and slow CLVs to become tangent along the dominant fast direction, breaking the transversality between fast and slow subspaces. Building on this cascade, we propose two precursors to forewarn the occurrence of extreme events. We numerically test the theory and precursors on low- and higher-dimensional systems. The proposed precursors predict extreme events and critical transitions with 100% precision and recall. This work opens opportunities for time-forecasting extreme events using theoretically grounded precursors.
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nlin.PS 2026-04-15

Reduced equation tracks wave number to predict Ginzburg-Landau singularities

Reduced wave number dynamics in the real and complex Ginzburg-Landau equations

Scalar model from WKB expansion yields exact shock profiles and instability criteria confirmed by simulations in real and nearly-real cases

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We study large-scale dynamics in the Ginzburg-Landau equation (GLE) using a reduced description derived from a WKB expansion. Rigorous mathematical results establishing that this reduced equation accurately approximates the full GLE are currently limited to the real GLE (RGLE) and exclude phase-slip dynamics. For the RGLE, we find that the reduced equation has conserved gradient form and show that, upon inclusion of a higher-order regularization, it admits exact stationary solutions. In the reduced dynamics, all nonuniform steady states are linearly unstable and among them, localized hole solutions identified through the reduced description differ from the classical hole solution of the RGLE due to Langer and Ambegaokar. In the Eckhaus-unstable regime, we derive a self-similar description of the approach to finite-time singularities in the reduced equation, with scaling exponents that agree with direct numerical simulations (DNS), and a similarity profile obtained from a nonlinear 4th-order boundary value problem. Extending the reduction to the complex GLE (CGLE) with nearly real coefficients introduces a Burgers nonlinearity that generates traveling shocks connecting two distinct plane-waves. We obtain exact expressions for the shock profile and perform extensive DNS to demonstrate convergence to the predicted profile in the appropriate large-scale, nearly real-coefficient limit of the CGLE. Away from this limit, the wave number profile loses monotonicity, which we explain in the framework of spatial dynamics. We further show that the exact shock solutions found here are qualitatively distinct from the Nozaki-Bekki solutions. Taken together, our results reveal how a single, scalar reduced equation elucidates unstable stationary states, self-similar collapse toward phase slips, and shock formation, providing an understanding large-scale phase dynamics in pattern-forming systems.
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nlin.CD 2026-04-15

Ensemble spread uncovers chaos features in lattices

Data-driven characterization of spatiotemporal chaos using ensemble reservoir computing

Uncertainty from multiple reservoir predictions marks frozen sites, tracks defect motion and measures turbulence levels in chaotic systems.

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Spatiotemporal chaotic systems are difficult to characterize in a model-free manner because of their high dimensionality, strong nonlinearity, and sensitivity to initial conditions. Coupled map lattices, as a representative class of extended nonlinear systems, exhibit diverse regimes such as frozen random pattern, defect chaotic diffusion, and fully developed turbulence. In this work, we propose an ensemble version of multiplexing local reservoir computing for the data-driven characterization of spatiotemporal chaos. By constructing multiple base learners with randomized hyperparameters and combining their outputs, the method improves prediction robustness and quantifies predictive uncertainty through ensemble spread. More importantly, we show that this uncertainty contains direct dynamical information. It identifies frozen positions in frozen random pattern, supports the estimation of defect diffusion coefficients in defect chaotic diffusion, and provides an effective indicator of chaotic intensity in fully developed turbulence. Analyses of the spatial power spectrum and Lyapunov exponent spectrum further support the consistency between the uncertainty field and the intrinsic dynamical properties of the system. These results show that ensemble reservoir computing can serve not only as a prediction tool but also as a data-driven framework for the dynamical characterization of high-dimensional nonlinear systems.
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nlin.AO 2026-04-15

One oscillator reveals coupling strength in large populations

Inferring coupling strength and natural frequency distribution in coupled Stuart-Landau oscillators using linear response

Natural frequency distribution is extracted from collective responses using linear response in Stuart-Landau systems.

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We propose a framework to infer the coupling strength and the natural frequency distribution in a coupled Stuart-Landau oscillator system with a large population. The inference method uses observation of linear response of a macroscopic quantity and of an oscillator. We first solve the direct problem on the response with transforming the system into the phase-amplitude equations. Solving the inverse problem, we show that the coupling strength is inferred from observation of an oscillator and the natural frequency distribution from macroscopic responses. The proposed method requires only one-dimensional observation in the two-dimensional Stuart-Landau system. Validity of the inference theory is examined by numerical simulations.
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nlin.AO 2026-04-15 Recognition

Single oscillator response reveals coupling strength

Inferring coupling strength and natural frequency distribution in coupled Stuart-Landau oscillators using linear response

Linear responses of one unit and the population recover coupling and frequency distribution in large Stuart-Landau groups

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We propose a framework to infer the coupling strength and the natural frequency distribution in a coupled Stuart-Landau oscillator system with a large population. The inference method uses observation of linear response of a macroscopic quantity and of an oscillator. We first solve the direct problem on the response with transforming the system into the phase-amplitude equations. Solving the inverse problem, we show that the coupling strength is inferred from observation of an oscillator and the natural frequency distribution from macroscopic responses. The proposed method requires only one-dimensional observation in the two-dimensional Stuart-Landau system. Validity of the inference theory is examined by numerical simulations.
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nlin.CD 2026-04-15

Chaos shapes quantum transport from single particles to many-body systems

Chaotic Dynamics and Quantum Transport

Review traces the role of chaotic dynamics across 40 years of theory development, from isolated particles to dissipative identical-particle,

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This chapter gives an overview of transport problems where chaotic dynamics of the system plays a crucial role. We begin with single-particle transport problems and then come to conservative and then dissipative systems of identical particles, which follows the historical way of developing the theory of Quantum Chaos over the past 40 years. We also include brief descriptions of key laboratory experiments on the discussed transport problems.
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