Papers

Canonical neural circuit motifs are usually described functionally: divisive normalization rescales population activity by a pooled signal, and winner-take-all competition selects one pattern through recurrent excitation and shared inhibition. We represent them, and their compositions, algebraically as finite transformation systems and analyze the transition monoids generated by their input-conditioned updates, distinguishing structure already present in a generator from structure that appears only through composition, and, on a joint state space, structure inherited from one factor from structure that lives on a joint configuration. Individually aperiodic updates can generate non-aperiodic monoids. In the WTA, every frozen-drive generator collapses to fixed points, yet short input sequences create local cycles of winner-dependent inhibitory gating: globally dissipative dynamics with a reversible action. The strongest result arises in WTA-to-DN composition. The composed monoid then contains a genuinely composite local cycle in which normalization state and the winner's gating state change together, although every primitive generator is aperiodic. Holonomy analysis certifies this as a group component of the Krohn-Rhodes cascade rather than an incidental cycle, and finds most group-carrying image sets on joint configurations, whereas the uncoupled product has none. An exhaustive interface sweep shows that the composite cycle is a property of the coupling rather than of a chosen map. If motifs are building blocks of neural computation, composing them is a form of programming: one chooses primitives and interfaces so that the generated algebra has the intended repertoire. The transition monoid is that repertoire - what a primitive presents to any later construction. Recurrent circuits are compositional transformation systems; their algebra constrains what they can be programmed to compute.

Disordered dynamical systems comprising many interacting units, from ecological communities to neural circuits, are ubiquitous, and understanding how connectivity shapes their collective behavior is a central theoretical challenge. One long-recognized feature of neural circuits is local connectivity balance, in which the excitatory and inhibitory weights converging onto each unit approximately cancel. Although local connectivity balance has been proposed to serve functions such as gating incoming signals, its effect on collective network dynamics remains unclear. Here we analytically study randomly connected recurrent networks with varying degrees of local connectivity balance. We show that this balance leaves the connectivity spectrum unchanged yet drastically reshapes the dynamics in a manner that depends critically on the single-unit nonlinearity. Local balance suppresses unbounded growth of the network state and stabilizes network dynamics when the activation function scales linearly or faster, whereas it drives the network into chaos when the activation function is sub-linear or saturating. Importantly, these effects vanish for odd activation functions, which are commonly assumed in previous work. We further find that, for saturating nonlinearities, the effective dimension of the dynamics varies nonmonotonically with the degree of balance. We show that all these phenomena arise from a unifying mechanism: the suppression of a self-generated feedback input by local connectivity balance. Our results identify local connectivity balance as a previously overlooked control parameter for collective dynamics in realistic disordered networks.

We study the vibrational dynamics that is responsible for Li-NC <-> Li-CN isomerization reaction using the Lagrangian betweenness, a tool recently developed to identify dynamical bottlenecks in the context of oceanographic transport. Following the spirit of the betweenness-centrality metric, which efficiently identifies the nodes that act as bottlenecks in complex networks, the Lagrangian betweenness is similarly capable to identify bottlenecks in dynamical systems. On the one hand, in this paper we show the ability of the Lagrangian betweenness to identify transition states, which are short-lived states that are formed at the top of the energetic barrier that the system must surpass in order to transition the reaction from the (initial) reactants to the (final) products. On the other hand, we also demonstrate that the Lagrangian betweenness is also able to identify other dynamical bottlenecks, and phase-space invariant structures such as manifolds, chains of islands and invariant tori, which are also responsible for the intricate vibrational dynamics of the system.

Synchronization is conventionally understood as the emergence of a phase-locked collective state among interacting dynamical units. However, extending this notion to systems whose dynamical variables are defined on higher-dimensional simplices introduces fundamentally new constraints arising from the topology of the underlying simplicial complex. In the simplicial Kuramoto model, nontrivial topological cycles give rise to a higher dimensional harmonic subspace that is unaffected by the coupling and can therefore drift indefinitely, preventing a globally synchronized state. To resolve this we employ Hodge decomposition on simplicial Kuramoto dynamics and investigated the components. Despite this topological obstruction, we show that the simplicial Kuramoto model admits fixed-point states in the exact and coexact sectors of the Hodge decomposition above critical coupling strengths, which we derive analytically for both sectors. This decomposition provides a generalized notion of synchronization in which the non-harmonic components converge to fixed states. We further investigate the robustness of these fixed-point states to external perturbations. Excluding the drifting and non-interacting harmonic component, the perturbation response is governed by the spectrum of weighted Laplacian and recovers a Kirchhoff index dependence analogous to standard Kuramoto case. In contrast, the harmonic sector is non-dissipative, and consequently the fragility of the system increases with the dimension of this topological subspace. We characterize this effect numerically using triangulated tori with varying first Betti number and find a superlinear scaling relation between system fragility and the dimension of the harmonic sector.

Nonlinear dynamical systems often evolve on low-dimensional manifolds with intricate geometry, making them difficult to approximate with a single global reduced-order model (ROM). To address this, we previously introduced quantized local reduced-order models (ql-ROMs), which partition the solution manifold into $K$ clusters and construct local ROMs around the corresponding centroids. In the sharp formulation, discontinuous switching between clusters can induce discontinuities, spurious oscillations, and loss of accuracy near cluster boundaries. Here, we propose a fuzzy extension of ql-ROMs based on fuzzy c-means clustering. The resulting fuzzy local ROM (fl-ROM) advances all local reduced states in parallel and combines their predictions through continuous membership weights, yielding a smooth partition-of-unity coupling of the local dynamics. We also introduce a pseudo-Bayesian information criterion to guide the selection of the number of clusters in the fuzzy setting. The methodology is assessed on the two-dimensional Kuramoto-Sivashinsky equation in periodic, travelling-wave-like, quasi-periodic, and chaotic regimes. In all cases, the local ROMs outperform the global ROM, while the fl-ROM provides smoother and generally more accurate predictions than the sharp formulation. In the chaotic regime, where long-time trajectory tracking is inherently limited, the fl-ROM achieves the best compromise between short-time predictive accuracy and long-term statistics. These results show that fl-ROM provides an effective, robust, and interpretable framework for reduced-order modeling of nonlinear dynamical systems with regime transitions and complex attractor geometry.

Leading eigenvalues of the truncated propagator, known as Ruelle-Pollicott (RP) resonances, are an elegant way of addressing the dynamics of unitary many-body systems. We study unitary propagators in their canonical form, known in the mathematical literature as the CMV matrices, and obtain a number of exact results for RP resonances and the associated norm-diverging eigenvectors. For the simplest CMV class describing a unilateral shift with an impurity, motivated by operator dynamics in dual-unitary circuits, we obtain closed-form results and in particular show that the three independent ways of obtaining RP resonances – the truncated propagator, analytic continuation of the resolvent, and the rigged Hilbert space approach – all give the same results. In more realistic CMV matrices, in which shift-like operator dynamics characteristic of chaotic systems is only asymptotic, we rely on the rich theory of orthogonal polynomials on the unit circle and identify two phases. In the first phase, relaxation occurs due to local operators effectively evolving into increasingly nonlocal ones with negligible backflow. Especially interesting is the second phase, which, surprisingly, exhibits faster relaxation because of contributions from the backflow of large operators. Additionally, in the second phase, RP resonances are not equal to the eigenvalues of the truncated propagator, instead, they are “hidden” within a ring of ill-conditioned eigenvalues.

This paper explores the use of a reduced manifold space initialization method for the generation of both time and fuel optimal many-revolution, low-thrust transfers between three-body periodic orbits within a family. Libration point orbit families are approximated by multi-segment Chebyshev polynomials and 1-dimensional Fourier series. Control is mapped into the family space, and a 1-dimensional targeting scheme in the family space dynamics is used to generate an initial guess for the phase space, directly accounting for winding. An integral collocation-based direct method is then used to first continue a feasible trajectory into the phase space, and then optimize for a particular cost function. The methodology is first applied to the distant retrograde orbit family. Modifications for time-regularized dynamics are then discussed, with an ensuing example of many-revolution transfers in the $L_2$ halo family.

We investigate bottleneck formation in turbulence using the Voigt-regularised SABRA model and DNS of the corresponding Voigt-Navier-Stokes (NSV) equations. The Voigt regularisation introduces a scale-dependent slowdown of nonlinear interactions without enhancing dissipation, providing a natural setting to study the interplay between nonlinear transfer and thermalised behaviour. We find three distinct spectral regimes: an inertial range at $k<k_I$, an intermediate equilibrium-like range associated with partial thermalisation for $k_I<k<k_{II}$, and a high-$k$ thermal regime for $k>k_{II}$, where the Voigt contribution dominates the conserved invariant. The crossover to the high-$k$ regime occurs at $k_{II}\sim 1/\alpha$, while $k_I$ marks the onset of thermalised behaviour. Equal and multi-time statistics reveal a progressive suppression of intermittency and a tendency towards Gaussianity at small scales, together with a transition from dynamic multiscaling in the turbulent regime to simple scaling in the equilibrium ranges. The shell model resolves these three regimes over a broad range of scales, while DNS of the corresponding NSV equations reproduces the same qualitative trends, including bottleneck formation, delayed cascade completion, reduced intermittency, and a tendency towards Gaussianity at small scales. We find that bottleneck formation might be associated with scale-dependent dynamical slowdown and incipient thermalisation, rather than being purely dissipative in origin. We provide strong evidence that, in the regime where the regularization parameter $\alpha$ is much smaller than the dissipation length scale, the Voigt model reproduces the same inertial-range turbulent regime and turbulence statistics as the Navier-Stokes (NS) equations. This provides evidence that the Voigt model constitutes an excellent practical approximation to the NS equations for small $\alpha$.

We present an open-source Python package for sampling the Gaussian, Laguerre, Jacobi, and Circular $\beta$-ensembles of random matrix theory. The package implements the Dumitriu–Edelman and Killip–Nenciu constructions, allowing efficient generation of random spectra for general $\beta > 0$. In addition to spectrum generation, it includes tools for the analysis of spectral statistics, from standard nearest-neighbor spacings and spacing ratios to non-adjacent $k$-spacings and the spectral form factor. These tools can be applied to generic spectral data, allowing users to compare them with and fit them to $\beta$-ensemble predictions. In this note, we review the $\beta$-ensembles, describe the package interface, and illustrate its use through several numerical experiments motivated by applications to quantum chaos. Our numerical results include an analysis of $\beta$ as a continuous fitting parameter in spacing ratio statistics, an examination of the numerical evidence for the conjectured $k$-spacing ratio distributions, and a study of the spectral form factor for general values of $\beta$.

Krylov state complexity, or spread complexity, has emerged as a sharp and versatile diagnostic of quantum chaos, information spreading, and many-body dynamics. Built from the Lanczos algorithm and grounded in the optimal-basis theorem, Krylov complexity thereby provides a robust spectroscopic window into quantum dynamics. A central theme is the characteristic overshoot observed in chaotic systems: a complexity peak in which chaotic evolution drives the state deeper into the Krylov chain than in integrable systems before relaxing to equilibrium. This behavior, tied to random-matrix universality classes of spectral statistics, is illustrated across a broad range of models, including quantum billiards, quantum spin chains, and variants of the SYK model. We also discuss proposed holographic descriptions of Krylov complexity in Einstein gravity, and conclude by outlining future directions and open problems, including time-dependent systems and quantum-field-theoretic formulations. A Mathematica notebook is provided for numerical exploration of Krylov complexity and spectral statistics across models.

Folding changes the linearized aeroelastic spectrum and can switch which mode becomes unstable first, with consequences for local postflutter interactions. This study formulates a three-component Z-fold wing by assigning every aerodynamic station to a structural component and a material coordinate. The same attachment map generates surface motion and returns pressure loads through virtual work. Geometrically exact component dynamics, an explicit-wake unsteady vortex-lattice model, and block-structured descriptor assembly preserve the physical paths of configuration actions. A two-parameter flutter analysis shows that a smooth flutter-speed envelope conceals a high-low-high sequence of controlling neutral branches, expressed as a critical-frequency valley and a redistribution of component deformation. Numerical continuation locates a near-1:2 double-Hopf point. Within a local model retaining quadratic and cubic structural restoring forces with aerodynamic and inertial operators fixed at the scheduling point, the cubic normal form captures selected 26-state observations and admits high-frequency-dominant and mixed phase-locked periodic solutions. The formulation links configuration-dependent flutter-mode identity to local resonant dynamics and supports blockwise sensitivity and design reasoning.

Forecast-error growth has been studied through complementary approaches: scalar models describe the evolution of bulk error, whereas spectral theories follow decorrelation and error transfer across scales. Their connection matters when a fitted growth curve is used to infer a physical mechanism or a limit of predictability. We establish a conditional link between these approaches. Within a bilinear spectral exchange model, assume a stationary reference spectrum and a remainder that vanishes on the proposed common-shape evolution. For a common initial decorrelation fraction strictly between zero and one, the spectral shape is preserved and its amplitude grows logistically exactly when every participating scale shares one positive growth coefficient. The converse inference fails: a logistic aggregate need not imply proportional decorrelation or a common coefficient. An exact two-shell construction demonstrates this failure with identical aggregate curves and different spectral dynamics. For heterogeneous amplitudes, an exact averaging identity separates the contributions of growth-rate mismatch, shape variance, rate–shape covariance, evolving spectral weights, and residual tendencies. A finite-time bound relates their combined effect to departure from a reference logistic curve. Forced surface-quasigeostrophic simulations illustrate how similar retrospective agreement with a logistic reference can coexist with heterogeneous effective shell rates and substantial instantaneous tendency discrepancies. Thus a scalar growth curve alone cannot identify how uncertainty is distributed across scales or justify interpreting a fitted rate as a common spectral growth coefficient. The framework specifies the additional spectral and budget evidence needed to give scalar predictability models that interpretation.

Using monthly Niño-3.4 anomalies through July 2026, we investigate how much predictive information is contained in delayed observations of the index. Ridge regression identifies informative delays, while multilayer perceptron and sparse identification of nonlinear dynamics (SINDy) models test whether nonlinear complexity provides additional direct forecast skill; gated recurrent unit (GRU) and long short-term memory (LSTM) networks provide a complementary test in which the temporal representation is learned internally. Delayed observations substantially improve forecasts over persistence and climatology at leads of up to six months, but increasing model complexity provides no systematic improvement. Historical recursive experiments favor a simple explicit SINDy recurrence and select shallow recurrent architectures, with no appreciable gain from learning the temporal representation internally. These results support a compact predictive representation of Niño-3.4 evolution in which the representation of past information is more consequential than model complexity. As a prospective application, the selected models are used to forecast the developing 2026 event beyond the last available observation and to compare its predicted evolution with completed historical El Niño events.

In oscillator networks sustained by finite resources, synchronization can depend on both the total resource and its spatial distribution. We study a duplex system whose activity layer consists of chaotic Hindmarsh–Rose oscillators and whose transport layer redistributes a conserved resource through a degree-biased Markov process. The stationary resource field is characterized analytically, and its existence, uniqueness, and convergence are established. By embedding this field into a local adaptive feedback law, the available resource is converted into node-dependent dissipation, for which Lyapunov analysis guarantees convergence to the synchronization manifold. Numerical results show that topology bias reorganizes the transient route to synchronization. Weak bias produces an almost collective contraction, whereas intermediate bias creates a hub-initiated recruitment hierarchy that extends toward middle-degree and peripheral nodes. Under stronger bias, the degree hierarchy becomes more pronounced while peripheral recruitment slows because adaptive dissipation is concentrated on structurally privileged nodes. Across the explored parameter range, this localization–coverage tradeoff is accompanied by a non-monotonic synchronization response at fixed total resource. The largest Lyapunov exponent remains positive after synchronization and the correlation dimension changes only modestly, consistent with suppression of transverse deviations while chaotic motion is retained on the synchronization manifold.

The Atlantic Meridional Overturning Circulation (AMOC) carries multi-decadal memory that ensemble-mean risk projections do not capture. We quantify this memory and its consequences for tail risk with a Volterra integral framework applied to ensemble-mean-subtracted variability in 14 CMIP6 models (1850-2100, three SSP scenarios), so that results reflect intrinsic thermohaline memory rather than the common anthropogenic forcing trend. Four results follow from leave-one-out cross-validation and an annealed-quenched tail decomposition. First, the ensemble-mean DFA1 Hurst exponent is H = 0.781 +/- 0.219, with 12 of 14 models showing long-range dependence (H > 0.5), consistent with thermohaline adjustment timescales near 33 yr. Second, a first-order Volterra model reduces out-of-sample RMSE by 16.8% over the best autoregressive baseline and by 12.3% over an unconstrained 20-lag distributed baseline, both robust to 200 circular-shift placebos (p < 0.05): the physically motivated kernel shape carries genuine predictive advantage. Third, the rolling 30-year lower-tail frequency rises 1.9-3.1x above the historical baseline depending on scenario, and the memory amplification index exceeds 1 in 8 of 14 models under SSP5-8.5 (median 1.15): persistence-driven clustering of weak-AMOC states amplifies tail frequency beyond forcing-only projections. Fourth, model-specific optimal memory horizons (15-52 yr) correlate with thermohaline regime (r = 0.74, p < 0.01), and the ensemble-mean Hurst exponent already exceeds a detection threshold H* = 0.70 in 9 of 14 models, with theoretical lead times to near-tipping conditions of 10-35 yr depending on scenario. These results support trajectory-specific, memory-aware AMOC risk assessment under moderate-to-high forcing and a physically grounded, scenario-conditional early-warning framework.

This work presents a new two-dimensional quantifier plane that combines normalized permutation entropy and normalized permutation mutual information, both derived from Shannon's information theory and normalized consistently through the embedding dimension of the Bandt and Pompe method. Each point on the plane simultaneously characterizes the intrinsic uncertainty of a variable and the amount of information it shares with another. We further show that a third quantity, equivalent to the conditional entropy of one variable given the other, introduces a notion of informational independence and, when both projections of a variable pair are considered jointly, reveals directionality between them. We analyze regular, chaotic, and stochastic dynamics to illustrate the relevance of the informational plane and show how these regimes evolve as a function of a coupling factor.

This work develops a discrete-time framework for identifying probability localization dynamics through finite stochastic representations adapted in space, time, memory, and state information. A compact dynamically relevant set is localized by a finite measurable partition, producing an observable probability state and a relational graph of admissible transitions. Structured stochastic liftings derived from Stochastically Structured Reservoir Computing (SSRC) give lossless polynomial representations of the observable state, while stochastic delay liftings add finite observable memory. These are distinguished from dynamically informed state-space enrichment: refinement of observational fibers containing states with the same present observation but different observable futures, yielding an exact obstruction-to-closure criterion. A route-network toy problem gives a minimal obstruction example, while four numerical laboratories (rotational phase dynamics, the chaotic logistic map, the Van der Pol oscillator, and a synthetic cyclic inventory system) show how spatial scale, temporal scale, polynomial degree, and delay depth interact. The logistic map isolates representation-induced memory in an otherwise Markovian chaotic system, using its exact invariant law as an ergodic benchmark and its zero-mass pseudospectrum to separate relaxation from transient amplification. An exact rotational cycle calibrates pseudospectra as a robustness diagnostic rather than a closure certificate. The inventory example gives a closure-driven enrichment procedure: residence-age hazards trigger age-refined states that improve predictive scores. These results motivate a minimal adequate representation: the least complex representation meeting predictive, structural, and identifiability requirements.

In a recent paper, a semiclassical Lyapunov exponent associated with a quantum Hamiltonian represented by a finite-dimensional Hermitian matrix was defined and placed on a mathematical foundation. The Lyapunov exponent characterizes the early stages of the evolution toward the ergodic state, while the late stages are characterized by the spectral gap of the corresponding Markov matrix. Here, we apply this formalism to five random-matrix ensembles. For each ensemble, we derive the mean Lyapunov exponent, its variance, and the spectral gap as functions of energy. We also present the corresponding thermal averages. Extensive numerical data are compared with the theoretical predictions.

The Bandt-Pompe permutation entropy framework, alongside the complexity-entropy causality plane, has become a standard tool for characterizing the dynamical properties of time series. However, observational noise distorts ordinal pattern probability distributions in ways that can systematically misplace time series within the causality plane, compromising dynamical classification. This effect is particularly relevant for geophysical signals, which are typically poorly and irregularly sampled, and have a low signal-to-noise level. In this work, we characterize the distortions on ordinal pattern statistics using the formalism of majorization. We provide theoretical results and propose corrective strategies that restore discriminability under realistic measurement conditions. To achieve this, we introduce methodology that allows the characterization of noisy dynamical series and further allows the quantification of observational noise without the need of a fitting procedure. Finally, we illustrate our methodology by analyzing paleomagnetic records to determine if the geological evolution of the Earth dipole is better described by a stochastic or chaotic system.

We consider a reduced Lorenz-like model that describes two walking droplets interacting through their mutual wave field, and investigate the emergence of strong bipartite correlations in this classical wave-particle system. The coupled nonlinear dynamics admit two invariant synchronization manifolds associated with correlated and anticorrelated states, within which the droplets display synchronized chaotic intermittency. Employing measurement protocols inspired by Bell experiments, we compute position correlations from the long-time dynamics and identify parameter regimes for which the CHSH-Bell parameter $S$ exceeds 2, corresponding to violations of Bell's Inequality. We further introduce a procedure for isolating the two subsystems, thereby ensuring the absence of wave-mediated signaling between them. Changing measurement settings following this isolation allows us to execute dynamic Bell tests in which violations persist. Our results demonstrate that nonlinear deterministic dynamics can produce Bell violations through wave-mediated synchronization mechanisms; moreover, these violations may be rationalized on the grounds that the wave form is influenced by the measurement settings. We thus provide a consistent dynamical framework for the appearance of classical entanglement in pilot-wave systems.

We investigate the propagation of counter-rotating convection rolls in Rayleigh-Bénard convection initiated locally in a quiescent fluid layer under supercritical conditions. The velocity of the front separating quiescent fluid from the forming convection rolls, and the wavenumber of the convection rolls remaining behind the front, are explored. We numerically investigate fronts of forming convection rolls over five orders of magnitude of the reduced Rayleigh number, $\epsilon$, in 2D and 3D domains, for a broad range of boundary conditions, and for different front initiation approaches. In all cases, the front velocity increases as $\epsilon^{1/2}$ with increasing $\epsilon$ for $\epsilon \lesssim 1$ in agreement with predictions using the amplitude equation. The amplitude equation description of the front velocity remains accurate for $\epsilon \lesssim 10$ except when the Prandtl number is large which yields a velocity that is faster than predicted for a fluid layer far from threshold. The wavenumber of the convection rolls increases linearly with $\epsilon$ in agreement with the wavenumber that maximizes the growth rate of perturbations in the linear regime. Farther from onset, the wavenumber growth transitions to a reduced scaling of $\epsilon^{1/4}$ in agreement with predictions using the Swift-Hohenberg equation in the large $\epsilon$ limit. The scalings describing the wavenumber variation with $\epsilon$ are independent of the domain geometry, boundary conditions, and front initiation method. However, the front-selected wavenumber at criticality does not equal the critical wavenumber of the bulk instability, in general, and depends significantly upon these details. We compare our results with experimental measurements where possible.

We study quantum dynamics through the lens of entanglement generation and characterize the underlying unitary evolution by the distinct signatures it imprints on the time-dependent entangling power. For a unitary operator, we characterize ergodicity by the equality between its long-time-averaged entangling power and the Haar-averaged linear entropy. We define mixing more stringently as the convergence of the time-dependent entangling power itself to the Haar value at long times. Within this framework, we establish the ergodic hierarchy of dynamical behavior, showing in particular that mixing implies ergodicity, whereas ergodicity does not necessarily imply mixing. As an application, we find that two-qubit unitary gates are neither ergodic nor mixing: their long-time-averaged entangling power can take only four discrete values, none of which coincides with the Haar average. We then investigate many-body dynamics using the kicked Ising chain and find that the long-time-averaged entangling power converges to the Haar value in both integrable and nonintegrable cases, indicating ergodicity. Remarkably, however, the nonintegrable chain exhibits mixing, whereas the integrable chain, despite being ergodic, is demonstrably nonmixing. We also introduce a Lyapunov-like exponent to characterize the rate at which the time-dependent entangling power approaches its saturation value. We find that this exponent increases systematically with the degree of integrability breaking in the many-body system. Our results establish entanglement generation as a useful framework for characterizing dynamical systems and reveal qualitatively different signatures of integrability beyond conventional diagnostics.

Strict thermodynamic scaling relations, such as the Rushbrooke inequality, are fundamentally established for equilibrium critical phenomena in the thermodynamic limit. In finite-size dynamical systems exhibiting synchronization, the direct application of such identities is hindered both by the finiteness of the network and the nonequilibrium nature of the spontaneous synchronization transition. To bypass this difficulty, we rigorously study the dynamical counterparts of the order parameter, susceptibility, and specific heat in finite systems of dynamical oscillators with nonlinear coupling. By measuring these quantities as a function of system size, we extract the associated critical exponents governing the transition. The validity of our thermodynamic mapping is tested by directly confirming the Rushbrooke inequality. Our results establish that standard equilibrium thermodynamic scaling architectures can be systematically applied to the finite-size scaling of nonequilibrium synchronization dynamics.

Nonlinear dynamical systems manifest rich variety of dynamical states and transitions driven by fluctuations. To understand the pattern of fluctuations during dynamical transitions, we investigate the structural features of chaos to order transition in logistic map. We determine fluctuations as amplitude jumps and encode them onto a complex network where nodes represent amplitude levels and links represent transitions between distinct amplitude bins. We discover that global network measures identify points of period doubling, regimes of periodicity and chaos, including interior crises events. Using local network measures, we also unravel novel peculiar parabolic-shaped patterns in the orbit diagram that we show are reminiscent of the distribution of stable and unstable periodic points in the bifurcation diagram.

Periodic surface-wave forcing can reorganise turbulent flows through coherent spectral response, synchronization, intermittency, and changes in short-term predictability, so its dynamical effect need not vary monotonically with forcing frequency. We reanalyse laboratory acoustic Doppler velocimetry records obtained at constant discharge under four conditions (0, 0.5, 0.67, and 1~Hz) using one common KNN–GMAE forecast-error-growth protocol. A distance-weighted $k$-nearest-neighbour predictor generates out-of-sample forecasts over multiple horizons, and the slope of the early quasi-linear region of $\ln(\mathrm{GMAE})$ versus physical forecast time is reported as a finite-horizon forecast-error-growth rate, $\lFEG$. For the full 120-s records, $\lFEG$ is 5.68, 0.85, 3.39, and 4.11~s$^{-1}$ for 0, 0.5, 0.67, and 1~Hz, respectively. The ordering 0~Hz $>$ 1~Hz $>$ 0.67~Hz $>$ 0.5~Hz is preserved in all seven nearby parameter configurations, indicating that the comparative result is not an artefact of a single KNN setting. A 40-s sliding-window analysis with a 10-s step reveals substantial temporal structure. The no-wave condition remains predominantly high and the 0.5-Hz condition predominantly low, whereas the 1-Hz record has weaker local support for a single exponential-growth regime: only 3 of 9 windows satisfy the adopted early-fit criterion $\RFEG\geq0.90$, compared with 8/9, 7/9, and 7/9 for 0, 0.5, and 0.67~Hz.

The Trachenko-Zaccone equation provides a compact nonlinear dynamical framework for describing non-Debye relaxation in disordered condensed matter. Originally developed to rationalize stretched- and compressed-exponential relaxation in liquids and glasses from the dynamics of interacting local relaxation events, the same equation has subsequently appeared in broader contexts, including polymer relaxation and nonlinear models of global population dynamics. This review retraces the conceptual development of the equation, with particular emphasis on its physical origin in Kostya Trachenko's feed-forward interaction mechanism, its mathematical structure, and its possible generalizations. We also include personal recollections of the work with Kostya Trachenko at Queen Mary University of London in August 2019, during which the equation emerged in essentially its present form. After reviewing applications to stress relaxation, glassy materials, polymer relaxation and population dynamics, we discuss future directions including time-dependent feedback parameters, coupled order parameters, heterogeneous and spatially resolved formulations, flux terms, network versions and stochastic extensions. The central theme of the review is that the Trachenko-Zaccone equation should be viewed not only as a model of glassy relaxation, but as a general nonlinear feedback equation with potential applications across complex systems.

Extreme events (EEs) in chaotic dynamics are rare broad excursions whose forecastability can be altered by dynamical noise. We investigate how noise changes EE occurrence and prediction skill across forecast horizons in a third-order autonomous chaotic flow. A single clean-data threshold is frozen for all realizations, broad events are defined by one maximum per excursion, and a future window W=15 is predicted from a 15-time-unit history using HistGradientBoosting with chronological data separation. As the forecast gap G between the observed history and the future event window increases, the clean Matthews correlation coefficient (MCC) decreases from 0.641 at G=0 to 0.165 at G=15. Noise dependence is evaluated with ten paired realizations at eight amplitudes. The mean short-horizon score increases from 0.456 in clean data to 0.546 at sigma=0.007; the paired gain is 0.0895 (95% CI 0.0494-0.1295; Holm-adjusted p=0.0234). Noise strongly increases EE occurrence while event amplitude and width remain comparatively stable. Equalizing positive training counts across noise levels substantially attenuates the short-horizon gain, whereas strong noise reduces intermediate-horizon skill. We term this horizon-dependent, nonuniform change in forecast skill noise-induced predictability redistribution (NIPR).

In the article a new approach to construction of generalized coherent states in quantum billiards is proposed. The coherent states are defined as the projections of a Gaussian wave function on the basis of the eigenstates of the quantum billiard, continued outside. The continuation is built as the solution of the equivalent Balian–Bloch equation, defined on the entire $\mathbb{R}^2$ and the projection operator is built using the resolvent of the Balian–Bloch equation. In the case of the one–dimensional potential well and billiards, belonging to the Coxeter group, the wave functions of the coherent states were expressed analytically via the Jacobi and Riemann theta functions.

The intermediate-energy double pendulum is a prototypical chaotic system. Despite its irregular motion, it exhibits recurrent flips - events in which one of the arms passes over the top. We develop a statistical prediction for the mean flip rate in terms of phase space flux. Applying this flux-based approach to the equal-mass, equal arm ("egalitarian") double pendulum, we find excellent agreement between the statistical prediction and numerical simulations: ensemble-averaged flip rates agree at about the 1% level, while even the statistics of individual chaotic trajectories agree at the few-percent level, quantifying the validity of the ergodic approximation. This agreement holds for flips of either arm and over a broad range of energies. This flux-based approach is closely related to that used for the egalitarian three-body system. A central role is played by the saddle orbits: periodic orbits that tend to the stable saddle eigenmode as the energy approaches the saddle energy from above and approximately follow the ridge of the potential at higher energies. These orbits provide a natural dividing surface for defining flips while avoiding recrossings. Indeed, the resulting distribution of crossing times exhibits a distinct gap. We also present two alternative simplified dividing surfaces, one of which is based on an accurate analytic approximation to the saddle orbits.

Weather forecasting and climate projection frequently use multi-model ensembles (MMEs) to improve short-term forecasts by averaging across models. However, this practice is often not well justified or validated. Using reservoir computing (RC) as a computationally efficient alternative to large-scale physical models, we assess the validity of the MME approach for chaotic time series. By training multiple randomly constructed RCs on the same dataset, we create a multi-model ensemble in which each model has its own unique error. These model errors lead to very different forecasting performances, with forecast error distributions that exhibit heavy tails. The arithmetic mean across forecasts from multiple models for the same target is usually closer to the ground truth than most individual forecasts, and further improvement is achieved by weighted arithmetic means where the weights are constructed based on each model's test-set performance. We show that iterated forecasts over many time steps deviate from the ground truth along the unstable manifold of the target point, in both directions, so that, if forecast errors were independent and had zero mean, the arithmetic mean forecast should approach the true target like $1/\sqrt{\nens}$ where $\nens$ is the size of the multi-model ensemble. We observe deviations from this behavior, which we attribute to the tails of the error distribution of random RCs.