Two-state stochastic models, where motion alternates between distinct dynamical modes, are widely observed in complex systems. Here we study the Two-State Random Walk (TSRW), which switches between a continuous-time random walk (CTRW) rest state and a standard L'evy walk (LW) motion state, each with power-law distributed sojourn times. Using anomalous diffusion decomposition, we show that TSRWs exhibit a generic coexistence of Joseph (correlation), Noah (heavy-tailed increments), and Moses (aging) effects. Strikingly, although classical L'evy walks alone possess only the Joseph effect, both Noah and Moses effects emerge in TSRWs solely due to stochastic switching with the CTRW phase. Our results demonstrate that coupling between dynamical states can fundamentally reshape the mechanisms driving anomalous diffusion, offering a minimal yet powerful framework for transport in heterogeneous and intermittently switching environments.

We consider integrable hierarchies such as KP, modified KP, 2D Toda lattice, BKP (small and large), DKP, Pfaff-Toda and their multi-component generalizations. We work in the framework of the bilinear formalism in which the universal dependent variable is a tau-function satisfying bilinear equations of the Hirota-Miwa type. Our principal interest in this paper is the dispersionless versions of the hierarchies. In the limit of zero dispersion the main object is an $F$-function, which is the limit of properly re-scaled logarithm of the tau-function. We show that in all the cases there exists an algebraic curve built into the structure of the hierarchy. We call it the {\it dynamical curve}. For the KP, modified KP and Toda lattice hierarchies, as well as for their multi-component generalizations, the curve is rational (of genus 0) and can be uniformized by rational or trigonometric functions. For hierarchies of the Pfaff type (DKP and Pfaff-Toda) the dynamical curve is in general a smooth elliptic curve (of genus 1), with its modular parameter being a dynamical variable. It is also shown that the large BKP hierarchy admits two different dispersionless versions. In one of them the dynamical curve degenerates to a rational curve while in the other one it remains to be elliptic. We show that a reformulation of the hierarchies based on uniformization of the dynamical curves by elliptic (or trigonometric) functions makes their structure nice and clear, especially in the multi-component case.

In the framework of the one-dimensional nonlinear Schrodinger equation (1D-NLSE) of the focusing type, the present paper studies numerically rogue waves (RWs) that emerge in the nonlinear stage of noise-induced modulational instability of a plane wave. For a large ensemble of simulations, all sufficiently large local maximums of the wavefield (i.e., RWs) are systematically collected and analyzed. It is shown that the frequency of occurrence of such maximums begins to increase from zero at the time, when the fourth-order moment of amplitude reaches its first (largest) local maximum, and then grows in an oscillatory manner approaching its asymptotic value at long time. Near the statistically stationary state, the 1D-NLSE generates a much larger number of RWs than a comparable linear system, but, in average, one RW affects a much smaller spatiotemporal area. The distribution of these RWs by maximum intensity represents an exponential-like function without noticeable changes in behavior, indicating a similar origin for RWs of significantly different amplitudes. The collected RWs are compared within a sufficiently large spatiotemporal window with nine exact solutions, of which two models reproduce RW dynamics much better than the others: a general collision of two solitons and a general collision of two Tajiri-Watanabe breathers; RWs are fitted with these collisions using pre-compiled databases of such solutions. The parameters of the two models turn out to be quite similar to the direct scattering transform spectrum of the whole wavefield, supporting a hypothesis that RWs may emerge due to synchronization of a few nonlinear modes (solitons or breathers) in presence of many other nonlinear modes of the wavefield.

Real complex systems often exhibit collective transitions emerging from interactions across many components. Classical stability theory describes such transitions in spectral space, where dynamics is organized by spatially extended global eigenmodes whose collective nature obscures direct association with individual physical components. Here, we show that structural disorder in empirical random networks can fundamentally alter this picture. These properties induce spectral localization, causing Laplacian modes to concentrate on small subsets of nodes and producing a mode--node correspondence in which collective dynamics becomes governed predominantly by the local behavior of a dominant node together with their effective coupling to the surrounding network. As a consequence, stability properties can be interpreted directly in node space rather than purely in spectral space. Exploiting this principle, we develop a node-resolved framework that predicts transition onsets, identifies the nodes responsible for emergent collective behavior, and restores interpretability in systems where classical modal theories fail. In heterogeneous reaction networks, the same mechanism gives rise to exotic collective states where different subsets of nodes develop distinct dynamical behaviors beyond those associated with homogeneous assumptions. Our results show that complex network structures naturally generate spectral localization, revealing the microscopic drivers of macroscopic dynamics.

Solitons in integrable systems exhibit a dual wave-particle character, yet their identification as individual objects becomes ambiguous during interactions, where they deform and delocalize. We develop a microscopic, field-based description that resolves this issue by introducing exact space-time fields of charge, current, and velocity derived from the inverse scattering transform (IST) of the Korteweg-de Vries (KdV) equation. This framework enables individual KdV solitons to be tracked throughout interactions, providing a quantitative description of their trajectories and deformation beyond the asymptotic regime. We show that the dynamics of $N$ interacting solitons can be formulated in terms of $N$ independent continuity equations, in which interactions are encoded in initial correlations that are subsequently propagated. From this microscopic viewpoint, effective velocities and hydrodynamic behavior emerge upon coarse graining, recovering kinetic theory of soliton gases and Generalized Hydrodynamics as scaling limits. Our results establish a direct connection between the wave-based IST formalism and particle-like emergent descriptions, offering a unified framework for soliton dynamics across scales.

We analyze a stochastic 5-neighbor cellular automaton with several conserved quantities, including the particle density. By examining the eigenvalue problem of the associated transition matrix, we derive an explicit formula for the stationary distribution on each irreducible component, in which the weight of each configuration is expressed in terms of the numbers of occurrences of two specific local patterns. This analysis further allows us to theoretically derive the dependence of the mean flux on the conserved quantities. In particular, we recover the mean flux formula in the deterministic case by taking the zero-noise limit of the system.

Failure theories often identify a single leading route to failure: the most unstable mode, weakest link, minimum-action escape path, or optimal perturbation. Yet finite-amplitude susceptibility depends not only on the nearest route but on how much of perturbation space lies near dangerous directions. We cast this distinction as a fragility problem: for each perturbation direction, the failure distance is the smallest amplitude that crosses a prescribed boundary, and the fragility curve is the fraction of directions that fail below a given amplitude. Measuring this curve directly requires nonlinear trials over many directions; instead, we show that it is predicted, before any failure occurs, by the tail of a single pre-failure quantity: the boundary-normalized fragility gain computed from the linearized response. The breadth of the associated response spectrum sets how many near-dangerous pathways coexist beyond the strongest direction. We demonstrate the mechanism in a high-dimensional nonlinear non-normal network with the strongest directional gain held fixed: the system with broader response-channel breadth has a larger nonlinear fragility curve, isolating breadth from the worst direction. An independent scalar test in deterministic traffic breakdown confirms the predicted sign: response breadth lowers calibrated jam thresholds once the strongest response is matched, with residual margins screening but never reversing the effect. Response-spectrum breadth thus emerges as a pre-failure coordinate for finite-amplitude fragility beyond the strongest path.

We consider natural Hamiltonian systems with potentials that are $C^0$ or $C^1$ on a hypersurface and $C^{\infty}$-smooth in the complement and introduce and study corresponding notions of their integrabilty and non-integrability. As a motivating example, we derive and analyze models of bungee jumping. We provide prototype examples of the Liuoville-Arnol'd theorem for $C^0$ and $C^1$ Hamiltonians.

From spin ice and crumpled paper to cold atoms lattices and metamaterials, geometrical frustration occurs generically whenever local constraints cannot be satisfied all at once. The result is a ground state degeneracy, where many equivalent states, each of which contains unsatisfied constraints, coexist. Here, we introduce dynamical frustration, where the ground state degeneracy makes way to a non-reciprocal self-oscillating state instead. To create dynamical frustration, we construct metamaterials that are driven parametrically in time and modulated in space. The parametric pumping leads to period doubling and in turn to a discrete symmetry-breaking. This symmetry breaking, together with the spatial modulation enforces the existence of topologically protected phase dislocations, which propagate unidirectionally with a spontaneous phase that breaks a continuous symmetry. Tesselating 1d frustrated loops, one obtains a 2d metamaterial where phase dislocations self-organize into globally synchronized non-reciprocal phase defects. We expect dynamical frustration to be broadly applicable at any scale, from cold atoms and superconducting circuits to acoustics and RF circuits -- anywhere where space-time modulation can be pushed beyond linear stability.

Collective intelligence emerges from local interactions among agents with limited information, yet how internal controller organization relates to emergent collective order remains unclear. Here, we study evolved swarms with shallow neural controllers under explicit sensory and actuation constraints and compare collective order with hidden-layer complexity and output nonlinearity across 3024 conditions. Under these constraints, the most ordered regimes exhibit two simultaneous and seemingly contrasting effects: hidden-layer complexity increases, while the effective output mapping becomes more linear. The diversity of recurrent collective behaviors varies nonmonotonically across the control parameters, with pattern richness shaped by parameter-specific tradeoffs rather than a single generic constraint optimum. Unevolved controls show that output linearization persists without adaptation, whereas the hidden-complexity relation depends on optimization. These two effects are respectively consistent with the law of requisite complexity and ecological rationality, suggesting that adaptive collective intelligence can arise through a partitioned controller organization in which representational complexity and action-level linearization coexist within the same system.

Geometrical frustration, a central paradigm in condensed matter physics, provides a unifying language for systems in which locally preferred interactions cannot be made globally compatible. Here, we use this language to formulate a minimal theory of frustrated neural timing, mapping repulsively coupled rhythmic units onto antiferromagnetic XY models. Within this framework, the condensed-matter concepts of local constraints, degenerate ground-state manifolds, metastability, and quench dynamics become a concrete diagnostic framework for structured neural phase dynamics. We analyze a hierarchy of geometries: a triangle as the minimal frustrated motif with two chiral 120° timing states, a tetrahedron whose reduced ground-state manifold consists of intersecting continuous branches associated with antipodal pairings, and a kagome lattice on which local constraints define a constrained three-coloring manifold. The kagome lattice reveals the central dynamical result: zero-temperature relaxation suppresses global synchrony but typically selects low-energy metastable torque-balanced states rather than exact ground states. Finally, we show how the phase theory can be carried back towards biophysical neural models by treating it as an effective-interaction target, where geometrical timing frustration is realized through preferred phase lags that become incompatible around closed motifs. This perspective suggests that weak global coherence in neural systems does not necessarily signal disordered activity, but can reflect structured local timing order shaped by a frustrated dynamical landscape.

The destruction of regular regions in two-dimensional, area-preserving maps is traditionally described in terms of the breakup of invariant curves and the persistence of transport barriers. Here, we investigate how this scenario changes when continuity is lost. We study the extended standard nontwist map with a perturbation whose period differs from a full revolution on the cylinder. In this setting, the induced map on the cylinder becomes discontinuous, even though the map remains smooth on the real line. Using complementary chaos diagnostics, we find that regular islands are not enclosed by a single invariant curve but instead undergo hierarchical fragmentation into smaller regular components connected by chaotic channels. We show that trajectories initialized near elliptic points exhibit long trapping followed by escape, ruling out the existence of a global transport barrier. The fragmentation occurs when island chains are centered on the discontinuity line, while island chains away from it preserve the conventional islands-around-islands structure. By restoring continuity of the induced map on the cylinder in a modified formulation, we recover smooth invariant curves and eliminate fragmentation, demonstrating that the hierarchical structure originates from discontinuity rather than twist violation alone. Similar behavior is also observed in other two-dimensional area-preserving maps, indicating that the phenomenon is not restricted to nontwist systems.

Low-dimensional reduction theories such as the Ott-Antonsen ansatz have played a crucial role in the study of populations of coupled oscillators. However, most of these theories apply only to models in which the interaction is described by a single harmonic component, limiting their use in more realistic oscillator models. Using the theory of orthogonal polynomials on the unit circle (OPUC), we develop a low-dimensional reduction theory for populations of globally coupled phase oscillators with multiharmonic coupling. We show theoretically and numerically that it is exact for uniformly rotating solutions and provides a good approximation for nonequilibrium solutions.

This text is also a program that computes the number of one-dimensional number-conserving cellular automata with radius 1. At the end of the text, the numbers of such automata with up to 7 cell states are shown.

We study the behavior of perturbations in a compressible one-dimensional inviscid gas with an ambient state consisting of constant pressure and periodically-varying density. We show through asymptotic analysis that long-wavelength perturbations approximately obey a system of dispersive nonlinear wave equations. Computational experiments demonstrate that solutions of the 1D Euler equations agree well with this dispersive model, with solutions consisting mainly of solitary waves. Shock formation seems to be avoided for moderate-amplitude initial data, while shock formation occurs for larger initial data. We investigate the threshold for transition between these behaviors, validating a previously-proposed criterion based on further computational experiments. These results support the existence of large-time non-breaking solutions to the 1D compressible Euler equations, as hypothesized in previous works.

The functioning of complex systems depends on the coordination of diverse components, often supported by regulatory structures that incur costs. In human organizations, such costs manifest as administrative burden, which has been rising despite often reducing efficiency. Classic explanations point to bureaucrat self-interest or regulation, yet they do not explain variation across organizations or clarify how this burden can be reduced. Here, we develop a dynamical model of administrative growth that integrates known behavioral mechanisms of process creation, obsolescence, and removal. The model conceptualizes processes as developed for problem solving, but becoming obsolete as conditions change, while continuing to consume resources until actively pruned. This interplay generates two long-term outcomes: stable equilibrium or run-away growth. The threshold separating these outcomes is shaped by organizations' propensity to create new processes when faced with problems, and their propensity to prune obsolete ones in response to administrative burden. Importantly, their effects are asymmetric: sufficiently high creation propensity leads to bloat regardless of pruning propensity. Faster environmental change shifts this threshold, making bloat more likely. Simulations of interventions show that lasting reductions in administrative costs and waste require permanent shifts in priorities and investments in distinguishing obsolete from useful processes. Temporary efforts or indiscriminate cuts provide only short-lived relief, and counterintuitively, prioritizing direct production can increase waste. Our work highlights a general mechanism by which well-intentioned problem-solving can create self-reinforcing inefficiencies in complex systems, offering insights possibly generalizable to broader applications, such as legal, policy, and software systems where obsolete elements accumulate.

In the context of energy storage at the nanoscale, exploring the notion of \textit{quantum advantage} implies walking on the thin line at the boundary between quantum mechanics and thermodynamics, which underpins our conventional understanding of battery devices. With no classical analogue, the Sachdev-Ye-Kitaev (SYK) model has emerged in the last years as a promising platform to boost charging and storage efficiency thanks to its highly-entangling dynamics. Here, we explore how the robustness of this setup by considering the sparse version of the SYK model, showing that, as long as chaos is not completely broken, reducing its complexity may lead to more efficient quantum batteries.

We show that the bias of the approximate maximum-likelihood estimation (MLE) objective of a persistent chain energy-based model (EBM) is precisely equal to the thermodynamic excess work of an overdamped Langevin dynamical system. We then answer the question of whether such a model can be trained with minimal excess work, that is, energy dissipation, in a finite amount of time. We find that a Gaussian energy function with constant variance can be trained with minimal excess work by controlling only the learning rate. This proves that it is possible to train a persistent chain EBM in a finite amount of time with minimal dissipation and also provides a lower bound on the energy required for the computation. We refer to such a learning process that minimizes the excess work as minimal-dissipation learning. We then provide a generalization of the optimal learning rate schedule to general potentials and find that it induces a natural gradient flow on the MLE objective, a well-known second-order optimization method.

We consider the problem of reconstruction of an $n\times n$ matrix with coefficients depending rationally on $x\in \mathbb P^1$ from the data of: (a) its characteristic polynomial and (b) a line bundle of degree $g+n-1$, with $g$ the geometric genus of the spectral curve, represented by a choice of $g+n+1$ points forming a (non-positive) divisor of the given degree. We thus provide a reconstruction formula that does not involve transcendental functions; this includes formulas for the spectral projectors and for the change of line bundle, thus integrating the isospectral flows. The formula is a single residue formula which depends rationally on the coordinates of the points involved, the coefficients of the spectral curve, and the position of the finite poles of $L$. We also discuss the canonical bi-differential associated with the Lax matrix and its relationship with other bi-differentials that appear in Topological Recursion and integrable systems.

We present a theoretical and numerical dynamical-systems analysis of tubular origami tessellations by identifying the inverse module number, $N^{-1}$, as a perturbation parameter within the framework of Kolmogorov--Arnold--Moser (KAM) theory. In the large-module limit ($N \to \infty$), we show that the conservative dynamics formally converges to an integrable map with a variational structure, whose generating function corresponds to the total discrete mean curvature. From the viewpoint of KAM theory, nonresonant invariant curves of the integrable limit are expected to persist for sufficiently large $N$. Consistent with this expectation, numerical computations with increasing $N$ show that large regions of the phase space are filled with structures that appear to be invariant curves. By adjusting mountain-valley fold assignments and fold lengths, the system can be transformed into a nontwist map that exhibits multiple zero frequencies. The frequency profile in the integrable limit and the persistence of invariant curves allow us to control the number and arrangement of stable folding regions appearing as coexisting elliptic islands. These islands provide a phase-space interpretation of distinct folding modes, separated by invariant curves that act as geometric barriers to continuous deformation and obstruct transitions without self-intersection. Finally, we analyze the expanding and contracting dynamics of the origami structure within the framework of conformally symplectic systems. By introducing a virtual auxiliary fold as a drift control mechanism, we numerically confirm the existence of stable quasi-periodic attractors.

We model the spatiotemporal dynamics of cellular protein concentrations relevant to cell polarity near membranes composed of different lipids. Therefore, we consider a three-variable continuum model for membrane-bound protein, cytosolic protein, and the local composition of a binary lipid membrane. The model contains two globally conserved quantities: the total protein content and the average fractions of the two lipid species. It combines a conserved reaction-diffusion model for the protein dynamics, undergoing an active phase separation, with a Cahn-Hilliard equation for lipid demixing. Linear stability analysis of the homogeneous steady state and direct numerical simulations show that the lipid dynamics undergoes classical phase separation, whereas the protein dynamics exhibits oscillatory phase separation for intermediate total protein contents, associated with a long-wavelength instability and traveling domains. In parameter regions where both instabilities are present, we find multiscale patterns with larger-scale traveling and rotating protein domains coexisting with smaller-scale stationary lipid domains. In this regime, traveling protein domains coexist with arrested coarsening of stationary lipid domains above a critical coupling. We further show that the main instabilities and phase diagram are well captured by an extension of a recently proposed conserved FitzHugh-Nagumo model for non-reciprocal pattern formation. The extended model consists of two non-reciprocally coupled Cahn-Hilliard equations with different interface tensions, reflecting the distinct physical properties of lipids and proteins. This also explains the observed asymmetry between static lipid patterns and traveling protein patterns.

This paper investigates the nonlinear dynamics and phase transitions in power packet network connected with routers, conceptualized as macroscopic information-ratchets. In the emerging paradigm of cyber-physical energy systems, the interplay between stochastic energy fluctuations and the thermodynamic cost of control information defines fundamental operational limits. We first formulate the dynamics of a single router using a Langevin framework, incorporating an exponential cost function for information acquisition. Our analysis reveals a discontinuous (first-order) phase transition, where the system adopts a strategic abandon of regulation as noise intensity exceeds a critical threshold $D_c$. This transition represents a fundamental information-barrier inherent to autonomous energy management. Here, we extend this model to network configurations, where multiple routers are linked through diffusive coupling, sharing energy between them. We demonstrate that the network topology and coupling strength significantly extend the bifurcation points, with collective resilient behaviors against local fluctuations. These results provide a rigorous mathematical basis for the design of future complex communication-energy network, suggesting that the stability of proposed systems is governed by the synergistic balance between physical energy flow and the thermodynamics of information exchange. It will serve to design future complex communication-energy networks, including internal energy management for autonomous robots.