Papers

Harnessing integrability at arbitrary interaction strength, we build long-lived and spatially localized soliton-like excitations in the 1D Bose gas with contact repulsion (Lieb-Liniger model). We show how these density profiles, built from selective superpositions of multi-hole eigenstates, display long lifetimes tunable through parameters like dispersion width and number of holes. We further illustrate a scaling protocol that in principle enables unbounded increase in lifetimes in the thermodynamic limit. Our results showcase classical-like excitations originating from unitary quantum dynamics, away from the semi-classical regime, and in regions where interactions cannot be treated perturbatively.

Collective behavior in animal groups is routinely described in cognitive language: fish schools sense gradients, swarms compute, colonies decide. These formulations often leave open what the collective is inferring, what computational object is represented, or how that representation arises from microscopic interactions. Here we develop that description quantitatively in a simulation based on the canonical model of collective sensing in fish schools (Berdahl et al., 2013), in which collective gradient sensing emerges from individuals that do not estimate gradients themselves: agents only modulate their speed in response to local light intensity and respond to simple social forces from neighbors. We show that these local rules admit a mesoscopic Bayesian reinterpretation at the level of the school, and hypothesize that the collective dynamics realize approximate inference on a generative model of how darkness gradients drive inter-individual speed differences. The resulting school-level inference process yields a posterior distribution over the local darkness gradient, whose statistics vary with environmental structure and relate systematically to collective motion and sensing performance. The school can therefore be described, in a Bayesian-mechanical sense, as a distributed inferential system without requiring individual agents to perform probabilistic belief updates themselves. This yields a quantitative, falsifiable account of collective sensing as emergent inference, and a concrete bridge from collective behavior to Bayesian mechanics.

Accelerating the prediction of long-term behavior in chaotic systems is crucial in scientific computing. However, existing methods rely on numerical solvers or autoregressive models that advance one small step at a time, which makes long horizons expensive. We instead view this problem as learning the system's time-invariant evolution operator, which jumps the state across a large time span in a single evaluation. To this end, we derive the consistency equations a time-invariant operator must satisfy, with differential and compositional objectives in physical time. These equations also connect the learned operator to the physics-prescribed instant dynamics, enabling physics embedding in consistency learning. Across five chaotic systems, we find that physics-distilled consistency makes both short-term trajectories and long-term statistics more accurate. The learned operator survives temporal extrapolation and requires one-tenth as many evaluations as autoregressive rollout, offering an efficient route to long-term simulation of chaotic dynamics.

We consider the interaction-round-a-face version of the nineteen-vertex model and derive a new quantum spin-$1$ chain. In the isotropic limit, the associated spin chain can be seen as three coupled Babujian-Taktahjan spin chains with different boundary twists. We compute the ground state short-distance correlation functions of the isotropic model in the thermodynamic limit by exploiting the shared integrable structure between the models.

Griffiths phases provide a mechanism for sustaining critical dynamics over extended parameter regions without fine-tuning to an isolated transition point. This behavior has been mainly associated with structural heterogeneity in complex networks. Here we show that quenched heterogeneity in local parameters can generate a chaotic Griffiths phase in a network of map-based neurons. We study globally coupled Chialvo maps with quenched disorder in a local neuronal parameter, thereby isolating intrinsic parameter disorder from topological disorder. As the coupling strength is varied, a chaotic Griffiths phase emerges between desynchronized and coherent chaotic dynamics, characterized by intermittent formation and breakup of neuronal clusters and large fluctuations in the fraction of clustered neurons. Within this regime, cluster sizes exhibit power-law distributions with coupling-dependent exponents, providing evidence of critical behavior over a finite parameter interval. The chaotic Griffiths phase is absent in the homogeneous limit and broadens progressively as the degree of parameter heterogeneity increases. We further introduce topological disorder through small-world connectivity and find that its combined action with local parameter heterogeneity substantially extends the chaotic Griffiths phase. In both globally coupled and small-world networks, the number of positive Lyapunov exponents displays anomalous scaling with system size. These results demonstrate that intrinsic dynamical heterogeneity constitutes an independent mechanism for generating chaotic Griffiths phases and that its interplay with structural disorder enhances extended criticality in neuronal networks.

We derive long-time asymptotics in different space-time regions for the Sawada-Kotera (SK) equation with admissible data involving continuous spectrum and finite discrete spectrum. Our result proves the soliton resolution conjecture, yields the asymptotics governed by the F-XVIII Painlevé transcendent, and establishes the asymptotic stability of \(N\)-soliton solutions. More importantly, this extends the long-time asymptotic theory to the coexistence of solitons and radiation, whereas previous results were restricted to the purely continuous spectrum. One key ingredient in our argument is a new meromorphic Riemann-Hilbert framework that incorporates the discrete spectrum through complete six-point pole orbits while preserving cofactor analyticity and the singular structure at the spectral origin. A second ingredient is the combination of an orbitwise, region-dependent pole reduction mechanism and the \(\bar{\partial}\)-nonlinear steepest descent method, which allows us to select the solitons contributing in each region and separate them from the continuous radiation. Our framework also resolves the coalescing stationary points at the singular spectral origin through the regular modified SK problem and the Miura map.

An extension with temporal modulation is presented of the canonical 2 + 1-dimensional solitonic Dym equation as originally introduced by Konopelchenko and Dubrovsky. Application is made of admitted Painleve' II symmetry reduction to derive exact solution to a class of associated 2 + 1-dimensional nonlinear moving boundary problems of generalized Stefan-type.

Does stable autonomous generation select a direction of time? This question matters both for learning models that autonomously generate prescribed dynamics and for designing controllers that make an invariant set attracting without changing its dynamics. Periodic motion and quasiperiodic torus rotations admit stable generation in either direction. For a chaotic attractor, however, reversing the equations reproduces its backward trajectories while turning attraction into repulsion. Can additional latent variables or autonomous feedback restore attraction without changing the reversed motion? We prove an obstruction to such exact generation by finite-dimensional autonomous systems with $C^2$ vector fields, compact attracting sets, and continuous readouts. The result applies to the classical Lorenz attractor and, more generally, to dynamics with an ergodic invariant measure of positive metric entropy whose intrinsic local stable fibers are totally disconnected on a set of positive measure. The proof connects this geometry to an entropy contradiction: a hypothetical attracting realization would force the output entropy to vanish, contrary to the prescribed chaotic dynamics. We use reservoir-computing examples to illustrate the difficulty of stable autonomous generation for time-reversed chaos: both time directions are closely tracked while the networks are driven by data, but only the forward example retains Lorenz geometry during autonomous generation. The theorem identifies a class of dynamics that cannot be learned for exact, stable autonomous generation by finite-dimensional smooth models satisfying the stated assumptions, regardless of the learning procedure or the number of latent variables.

Inferring an Allen-Cahn constitutive force from images requires more than an accurate optimizer: nonlinear observation noise alters the mean of force features, shared noise couples instruments to the response, and a temporal balance borrowed from another solver can remain biased even on clean data. We develop observation-aware, noise-adjusted moment equations that address these effects before inversion. The surface-tension-calibrated constitutive transformer (SCCT) couples the resulting moments to a positive Bernstein inverse and can adapt its prior and precision from images and moments. A conditional stability bound separates moment residuals from prior error. Synthetic tests show that correcting the estimating equation matters more than increasing regularizer flexibility; the benefit of learned adaptation is smaller and depends on the training comparison. Forces inferred from planar movies predict unseen planar and three-dimensional evolutions. A ring-cage breakup shows why diffuse-interface accuracy distinguishes forces even when their topological predictions agree, while external surface tension sets the energy scale separately.

External potentials break the translational invariance of nonlinear Schrödinger (NLS) equations and generally convert a continuous family of standing wave solutions into isolated pinned states. We study this mechanism for the prototypical case of a kink in the NLS equation with competing focusing cubic and defocusing quintic nonlinearities. The Lyapunov–Schmidt reduction method is used to prove that the location $s_0$ of the pinned kinks is selected near roots of an effective potential $V_{\rm eff}(s)$. We then determine how the translational zero eigenvalue of the Jacobian operator splits under the perturbation and prove that the sign of $V_{\rm eff}'(s_0)$ distinguishes eigendirections of positive and negative energy. We further show that if the Jacobian operator admits a negative eigenvalue, then the stability problem contains a real unstable eigenvalue. The derivation of the asymptotic expansion for the unstable eigenvalue is nonstandard because the zero eigenvalue is embedded in the continuous spectrum of the linearized operator. Finally, we show that the persistence of kinks lacks parity symmetry even if the external potential is odd. Numerical continuation, spectral computations with domains adapted to the slowly decaying eigenfunctions, and direct dynamical simulations of the time evolution corroborate the analytical predictions for different representative examples.

We model directed motion of cells that aggregate into a cluster in the presence of chemotaxis or fluid transport. Starting from a stochastic agent-based model (ABM) of Galante & Levy (2013), Chavy-Waddy & Kolokolnikov (2016), we consider two model variants, with (I) directed motion of individuals, or (II) implicit individual bias in one direction. In both cases, we derive fourth order nonlinear PDEs representing the continuum limits of the models as a generalization of Chavy-Waddy & Kolokolnikov (2016). An exact analytic solution then describes cluster density profile as well as the group velocity, and we identify the regimes where the travelling cluster exists. Excellent agreement is observed between the discrete models and the exact solutions of the PDEs in both models, including the existence and speed of the travelling cluster.

This work establishes finite-time blow-up for the completely integrable defocusing Ablowitz-Ladik equation on both infinite and finite lattices. We first establish local well-posedness for arbitrary initial data in the space $\ell^\infty$ of uniformly bounded sequences, with no decay assumptions and under general boundary conditions. We then show that the unit circle acts as a separatrix for the lattice dynamics: sites with an initial modulus inside, on, or outside the unit circle maintain their modulus within their respective region for as long as the solution exists. As a consequence, initial data in the closure of the $\ell^\infty$ unit ball generate global-in-time solutions. On the other hand, the unit circle separatrix opens up the possibility for finite-time blow-up, which is restricted to sites with initial modulus outside the unit circle. We establish this scenario by constructing a class of initial data whose corresponding solutions are singular, with explicitly given exact blow-up times. We also construct solutions emanating from initial data with an arbitrarily large single-site excitation outside the unit circle that exist globally in time, a behavior driven by the convergence of the corresponding modulus to the unit circle equilibrium. In the presence of such solutions, along with other important classes of analytical, global-in-time solutions that remain outside the closed $\ell^\infty$ unit ball for all times, we examine the fundamental question of the dominant dynamics for general initial data outside the closed unit ball. By utilizing the constructed class of singular solutions as subsolutions for the general problem, we prove that the dynamics for the completely integrable defocusing Ablowitz-Ladik lattice is dominated by finite-time blow-up. We conclude with numerical studies of illustrative examples of general initial data

We construct spectral-parameter solutions of the quantum Yang–Baxter equation from the associativity condition of the universal symmetric $2$-algebraic two-valued group, whose multiplication law depends on four coefficients $k_2,k_4,k_6,k_8$ subject to the single relation $4k_8-k_4^2+k_2k_6=0$ – which coincides with the three-dimensional WDVV equation for Dubrovin's normalised potential. Every three-dimensional Frobenius manifold with a cyclic vector field carries such a spectral two-valued group, whose finite branch points are the eigenvalues of the multiplication operator. For an arbitrary unital algebra we show that a two-parameter spectral $R$-operator built from multiplication and unit has Yang–Baxter defect proportional to an explicit tensor built from the associator; for tangent Frobenius algebras this yields a regular, unitary $9\times9$ $R$-matrix satisfying the quantum Yang–Baxter equation exactly at WDVV points, with spectral parameter the Abel coordinate of a bielliptic curve whose elliptic quotient is the spectral curve, together with an associated integrable chain. The $A_3$ Frobenius manifold is worked out explicitly.

Selective advantage leads to the proliferation of a species into an ecosystem. On the other hand, self-propulsion can lead to dynamic clustering of individuals. To analyze the intricate effects produced by such competing processes, we introduce a minimal model for a proliferating active population, where run-and-tumble migration is combined with a stochastic birth-death process. Beginning with the microscopic dynamics of the particles, we derive the corresponding hydrodynamic equations and validate them through numerical simulations. We demonstrate that the inclusion of activity leads to the emergence of novel morphological patterns, which are sensitive to the specific rules governing migration. Furthermore, we show that activity significantly alters the characteristics of invasion waves, increasing their propagation speed. However, beyond a threshold, excessive activity disrupts the wave-like nature of the invasion process. We also extend our study to two dimensions and analyze the instabilities that lead to pattern formation in such systems.

We study the focusing one-dimensional cubic nonlinear Schrödinger / Gross–Pitaevskii equation (NLS-GP), where the potential of the underlying linear Schrödinger operator, $H_V=-\partial_x^2+V(x)$, is compactly supported. In \cite{turner2026resonance} the authors proved that purely imaginary zeros (transmission resonances) or purely imaginary poles in the lower half plane (scattering resonances) of a reflection coefficient of $H_V$ seed branches of resonance-induced nonlinear bound states, and that these states bifurcate at a strictly positive $L^2(\mathbb R)$ excitation threshold. This is in contrast to nonlinear bound states which bifurcate at zero $L^2$-norm from point spectra of $H_V$ (corresponding to poles in the upper half plane). In this paper we establish precise criteria for the nonlinear orbital stability and instability of the resonance-induced states near the bifurcation point. A corollary is that resonance-induced nonlinear states are orbitally stable if i) the underlying linear (scattering or transmission) resonance frequency is sufficiently small, and ii) the corresponding linear resonance eigenmode is strictly positive. Numerical simulations are presented to explore regimes not accessible to our theory, and to explore the large time dynamics for initial conditions near stable and unstable resonance-induced nonlinear bound states.

The periodically driven Duffing–Holmes oscillator is strongly multistable: for the same forcing amplitude and frequency, periodic, chaotic, and antiperiodic attractors coexist, each with its own basin of attraction. Antiperiodic orbits are periodic orbits invariant under a shift of half a driving period combined with a sign reversal of the coordinates, a symmetry of the equations of motion. Guided by a continuation map of the parameter plane, which locates the regimes but is blind to their coexistence, we explore the space of initial conditions along two one-parameter cuts, at fixed forcing frequency and at fixed forcing amplitude. Stochastic sampling of the full space yields an unbiased estimate of the relative volume of each basin as a function of the control parameter, on the same parameter grids on which single-trajectory bifurcation diagrams are computed by continuation. Basin maps on two-dimensional sections, complemented by maps of the order of the attractor reached from each initial condition, resolve the geometry of the coexisting basins where the competition between attractors is strongest. The relative volumes reveal windows of global antiperiodic dominance, in which the antiperiodic orbit attracts essentially every initial condition, separated by narrow intervals in which two or three qualitatively different attractors share the space in comparable proportions, as well as extended ranges of persistent coexistence; at large forcing amplitude, a single antiperiodic orbit of order one captures the whole space. Compared with the bifurcation diagrams, the abrupt changes of basin volume are the ensemble-level counterpart of the multistability-induced discontinuities of the sweeps, and the gradual shrinking of the antiperiodic basin explains why the branch followed by a sweep depends on the initial state and on the drive phase.

In an earlier paper of P. Rossi and the first author, a construction of a wide class of quantum hierarchies was presented, with the input data given by an arbitrary cohomological field theory. A crucial role in the construction is played by the double ramification (DR) cycles on the moduli spaces of stable curves. As a special case, one gets a quantization of the hierarchy of the Intermediate Long Wave (ILW) equation. The ILW equation has two famous limits: the Korteweg–de Vries equation and the Benjamin–Ono (BO) equation, and for the latter one a quantum hierarchy was constructed by Nazarov and Sklyanin. In our paper, we propose a conjecture describing a relation between a limit of the quantum ILW hierarchy and the Nazarov–Sklyanin hierarchy and present various checks to support it.

We investigate how root allometry and below-ground root competition due to root overlap affect the conditions for pattern formation in non-local scalar vegetation models. We start from a spatially explicit two-component vegetation-water system in which plants acquire soil water through their laterally extended root systems, with root allometry represented by an interaction kernel whose spatial extent increases with the above-ground biomass density. In the limit of fast water dynamics, we eliminate the water variable, resulting in a non-local scalar model for the vegetation dynamics. This reduction reveals that accounting for root overlap naturally leads to a double integral structure in the scalar model, in contrast to the single convolution term commonly used to model non-local competition for resources. For the resulting scalar model, we identify two distinct mechanisms that can drive Turing instabilities. The first instability mechanism requires root growth to outpace mortality, while the second mechanism relies on the development of competitive pressure zones, in which vegetation growth is impeded by the cumulative competitive pressure exerted by neighbouring areas with higher biomass density. Root allometry promotes the occurrence of both instability mechanisms, while root overlap inhibits the emergence of competitive pressure zones. Hence, our results show that both root allometry and root overlap are important features to consider when studying vegetation pattern formation.

In this paper we explore and extend the state of the art for understanding the skill and utility of early warnings of critical transitions (tipping points) in forced nonlinear systems. We highlight some of the challenges and opportunities of transforming estimates of a dynamics-based early warning system, based on trends in stability or resilience into forecasts with high skill. We highlight the importance of (a) quantifying consequences of possible actions in response to warnings that may be false negatives or positives (b) considering finite time horizon predictions to give verifiable predictions (c) assumptions necessary for valid extrapolations of trends. We evaluate approaches to improving the skill and utility of the forecast.

We examine whether the perturbative renormalization group (RG) regime of stochastic turbulence can be continued from small $\varepsilon$ to the physical value $\varepsilon_{\mathrm{phys}}=2$, corresponding to large-scale forcing. Exploiting the simplifications of the large-$d$ limit, we calculate the correction exponent $\omega$ to five-loop order and find a simple structure suggesting a closed analytic form. This form implies that the fixed point stable near $\varepsilon=0$ ceases to be simultaneously real and infrared stable before reaching $\varepsilon_{\mathrm{phys}}=2$, pointing to a possible change of infrared regime rather than a controlled perturbative RG extrapolation.

Understanding a nonlinear dynamical system from time series requires not only reproducing its trajectories, but also identifying a simple representation that preserves its essential dynamical structure. Almost-linear recurrent neural networks (AL-RNNs) are piecewise-linear RNNs in which only a subset of units use ReLU nonlinearities, so that nonlinear capacity is explicitly controlled by the number of ReLU units. Their activation patterns define linear regions, represented as symbols, whose observed transitions form a symbolic transition graph. However, directly training AL-RNNs with few ReLU units to realize minimal dynamical representations can be unreliable. We ask whether an AL-RNN with more ReLU units can instead be trained first and systematically reduced to a minimal dynamical representation. We introduce a fixed-point-guided hierarchical reduction procedure that progressively linearizes selected ReLU units, merging neighboring linear regions and graph nodes while preserving distinct symbols containing fixed points (FPs). The resulting reduction tree defines a hierarchy of progressively simpler candidates. Each reduced candidate is initialized from the parent parameters and retrained under guidance from the parent dynamics. We also prove that reproducing $Q$ distinct fixed points requires at least $Q$ FP-containing symbols, providing a certificate of symbol-level minimality when this bound is attained. On the 3-scroll Chua system, direct training with the theoretical minimum of three ReLU units achieves high-fidelity minimal realizations in only 20% of seeds, whereas our learn-reduce-retrain strategy increases the seed-macro success rate to approximately 71% at the same final nonlinear capacity. These results show that redundant nonlinear capacity can serve as a scaffold for discovering and realizing minimal dynamical representations.

Every scalar ordinary differential equation of order at least three with rational right-hand side that is locally linearizable by a point transformation admits a linear form with rational coefficients over the same coefficient field. We prove this by restricting the derived symmetry algebra to a coordinate line and recovering a scalar differential operator from rational symmetry-jet data. The construction uses differential elimination and linear algebra; it does not require the symmetry generators or a linearizing transformation to be solved for. A single integer parameter suffices to choose the line.

We demonstrate that two-component photonic nonlinear lattices with opposite signs of the out-of-phase spatially modulated SPM and XPM coefficients can support two types of optical vector solitons (VSs), the fundamental and mixed ones. The fundamental VSs are composed of two Gaussian-like components, while the mixed ones combine Gaussian-like and dipole components. The structures and stability of both types of the VSs are investigated. The VSs are stable, except for narrow intervals at small values of the propagation constant.

In this paper, we explore the impulsive dynamics common to single-joint, two-link models of walking and brachiating gaits with respect to slope and switching time. In particular, we investigate how the stability of a gait and bifurcations encountered within a family of gaits change under time-based and state-based switching of the impulsive dynamics.

We study a real-valued max-plus particle system on a finite periodic lattice. Its three-dimensional fundamental diagram relates the mean flow to two densities: the conserved particle density and a second density defined by a local state function. We prove that the deviation from the diagram tends to zero for arbitrary initial data with all state values in [0,1]. The proof uses two nonnegative local quantities and their evolution relations. For rational initial data with all state values in $[0,1]$, the orbit is eventually periodic, and the fundamental-diagram relation holds exactly on the periodic orbit.

This paper studies the focusing Calogero–Moser derivative nonlinear Schrödinger (CMdNLS) equation under the nonzero boundary condition \(|u(t,x)|\to 1\) as \(|x|\to+\infty\). In sharp contrast to the classical paradigm for the cubic nonlinear Schrödinger equation, namely that focusing nonlinearities support only bright solitons on a zero background, we show that the focusing CMdNLS equation admits a rich family of dark multi-soliton solutions on a nonzero background, whose existence is driven by a spectral gap \([-1,0]\) in the point spectrum of the Lax operator, together with the Hardy space structure. The main results are threefold. First, we establish an explicit formula for general solutions in the Hardy–Zhidkov space \(\mathcal Z^2_+\), via a semigroup method based on new commutator estimates between the Lax–Beurling shift semigroup and Toeplitz operators. This approach circumvents the non-commutativity of the unbounded generator with the time derivative and requires no weighted hypothesis on solutions or on the domain of validity of commutator formulas. Second, the algebraic definition of dark multi-soliton potentials is shown to be equivalent to two spectral conditions on the Lax operator. Together with the conservation of a generating functional, this yields the invariance of the dark \(N\)-soliton set \(\mathcal U_N\) under the CMdNLS flow. Third, the action-angle variables on \(\mathcal U_N\) are constructed, which establishes the complete integrability of the focusing CMdNLS equation on \(\mathcal U_N\). Combining these results, we obtain an inverse spectral formula, from which it follows that the poles of dark multi-soliton solutions satisfy a complexified Calogero–Moser system. Long time asymptotics and uniform Sobolev estimates are also derived.

Magnetic field-line transport in magnetically confined plasmas is commonly modeled as diffusion through stochastic magnetic layers generated by resonant perturbations. However, experimentally relevant magnetic configurations frequently exhibit transport rates that are substantially slower than predicted by fully stochastic models, indicating the presence of unresolved dynamical barriers. In this work, we investigate the role of nested resonant island chains and their associated invariant manifolds in regulating transport within the edge magnetic topology of DIII-D tokamak discharge #171491 subjected to strong n=3 resonant magnetic perturbations. Using field-line tracing, Poincare analysis, and manifold calculations, we identify a hierarchy of homoclinic and heteroclinic tangles associated with both period-one and period-ten hyperbolic points. A systematic construction based on primary intersection points is developed to extract partial transport barriers from the full manifold geometry and divide the chaotic region into dynamically distinct domains. Monte Carlo simulations reveal that field-line escape exhibits a bi-exponential character when only the outermost period-one boundary is considered, reflecting the presence of a long-lived "sticky" region associated with the period-ten island chain. When the inner period-ten boundary is explicitly incorporated into the Monte Carlo, the escape process becomes approximately single exponential and is accurately described by turnstile lobe transport. The measured escape rate agrees with predictions obtained from the turnstile lobe area and the symplectic structure of the field-line map. These results demonstrate that nested tangles provide a quantitative transport skeleton governing magnetic-field-line escape and establish a framework for analyzing transport barriers, cantori, and lobe dynamics in realistic fusion-plasma magnetic configurations.

The parity problem in 1D Cellular Automata is a classifier based on a local rule. If the number of ones in a ring of binary cells is odd then the automaton shall converge to all-one (11...11) else to all-zero (00...00). The objective is to find rules that are related to Fatès's Balanced Rule F but use less active neighborhoods (actions). The method is to analyze the effect of each actions and to find a functioning subset of the 12 actions used in rule F. New rules called HA, HB, HC, HF are defined with 10, 6, 4, 8 actions. Rule HC and HF need only three inputs whereas all others need four. The classification time of the rules was determined by averaging over many simulation runs with random initial configurations. It is almost the same for the rules HA and F, and is longer for the rules HB, HC, HF. In addition, the rule HD is defined, which updates three cells in parallel and works faster. In future work, the functioning and performance of the proposed rules should be treated in a more formal way, and related rules (also with different update schemes) could be considered.

There have been persistent efforts to uncover low-dimensional macroscopic dynamics in large systems of interacting microscopic units, in which underlying physical principles govern their collapse onto low-dimensional manifolds. Here, focusing on interacting phase oscillator systems, we show that an unsupervised autoencoder framework can determine their intrinsic dimensionality, in agreement with the predictions of Watanabe-Strogatz theory. Systematic sweeps over the latent dimensions identify the analytically predicted reduced dimensions, while supervised regression confirms that the learned latent variables capture the full information contained in the theoretical Watanabe-Strogatz coordinates. We demonstrate this capability for two prototype systems: Kuramoto oscillators and resistively loaded Josephson junctions. We further establish the diagnostic specificity of the framework by showing that systems lacking such reducible dynamics do not exhibit a sharp dimensional reduction.

Here, I give a brief history of the theory, with twists and turns, and list possible predictions for the spectra of the direct cascade of gravity waves on the water. The description of long waves is straightforward within the weak-turbulence theory developed by Zakharov and his school. As the energy cascade proceeds to shorter scales, the mean surface steepness grows, and wave-breaking events appear with increasing frequency. Considering surface cusps, Phillips suggested a wavenumber spectrum (with a wrong magnitude, as I argue here) and a frequency spectrum whose whole form is incorrect. The right form for the frequency spectral slope due to breaking waves was suggested by Kuznetsov. Here we estimate the magnitude and argue that the frequency spectrum is expected to continue unchanged through the whole cascade. The transition from weak to strong turbulence in the wavenumber and spatio-temporal spectra is due to spectrally nonlocal interactions and happens earlier (at larger scales) than the dimensional estimate suggests. As a result, the magnitude of the strong-turbulence spectrum is not universal but depends on the pumping for a sufficiently long weak-turbulence cascade. The reason for nonlocality is likely that breaking of shorter waves is determined by the magnitude of the longest wave on whose crest they are running.