Papers

The way numbers are represented strongly influences which arithmetic structures are easy to see. The prime clockwork is a recursively growing discrete dynamical system: a list of autonomous two-hand clocks driven by one common $+1$ signal. No primes or primality labels are supplied. Starting empty, the process appends a clock of period $n$ whenever none already present rings; the primes are generated internally as its growth times. For each installed prime $p$, the seconds reading $R_p$ advances through $0,\ldots,p-1$, and each return to zero increments the minutes reading $M_p$, which counts completed $p$-cycles. The hands use only increment, comparison, reset, and carry, without explicit mod or div operations. At time $n$, $n=pM_p(n)+R_p(n)$. The valuation readout $V_p(n)=\nu_p(n)$ is generated locally: it is zero when the seconds counter is non-zero (silent state) and otherwise (when the p-clock rings) one plus the earlier valuation addressed by the current minutes reading. The valuation vector gives the integer in unique prime-factorized form. Its coordinates add and subtract under multiplication and division, representing every positive rational uniquely; divisibility becomes weak componentwise order, and unique factorization is natural in this representation. Finite seconds arrays form Cartesian-product state spaces whose common orbit visits every joint state once before repeating; this grand cycle is the order-sensitive dynamical counterpart of the Chinese remainder theorem. The same coordinates expose gcd, lcm, perfect powers, Bézout's identity, and Euler's totient. Rational valuation levels reach certain positive algebraic irrationalities, but not algebraic numbers in general.

From protein complexes to electronic circuits, many natural and engineered systems function only if assembled in an exact, reproducible fashion. The structure of each of these systems can be understood as a network, yet network science lacks the mechanisms to consistently reproduce exact topologies, focusing instead on generating network ensembles. We introduce the framework of network design where we encode the local constraints obeyed by a system's building blocks in a design set, and derive the Unigraphical Design Theorem, which determines when these constraints guarantee reproducible assembly into a unique structure, a process we call unigraphical assembly. For systems whose design sets do not specify a unique outcome, we identify guided assembly as a second route to reproducibility, in which temporal ordering decomposes construction into unigraphical steps. Applying these results to 3,618 reproducible systems, including protein complexes, molecules, and robots, we classify those that undergo unigraphical assembly and those that require guided assembly. We further identify a diversity-redundancy boundary that explains how systems trade component variety for structurally interchangeable parts while retaining unique assembly. Finally, we experimentally test the theory using 3D-printed components to re-engineer generative construction sets into systems that assemble unigraphically into prescribed topologies. Network design thus reframes reproducibility as a mathematically testable property of real networks, opening a route to the rational engineering of complex systems.

Large-language models (LLMs) are rapidly becoming part of human culture, reshaping how information is produced, transmitted, and used. Here we propose that their diffusion can be understood through a viral analogy, with LLM use spreading through populations, becoming embedded in cognitive and cultural practices. We model transitions among uncoupled, coupled, and persistently dependent users, and show that the interplay between social transmission, recovery, and collective reinforcement can generate tipping points and technological lock-in. A central consequence is the possibility of runaway dynamics: once a critical threshold is crossed, small increases in adoption can trigger rapid population-level shifts toward persistent dependence, with abrupt losses in cognitive competence. The same framework, however, identifies conditions for cognitive immunization, based on reducing transmission and facilitating reversibility. Our results highlight how LLM adoption may involve nonlinear collective transitions with important consequences for cognitive autonomy.

The propagation of narrowband wave packets is a classical problem with broad relevance across nonlinear wave physics. Within the framework of the nonlinear Schrödinger equation, solitons are known to propagate without changing shape, whereas Gaussian envelopes may undergo substantial reshaping depending on the balance between dispersion and nonlinearity. Here, we combine laboratory experiments with numerical simulations to systematically investigate the evolution of hydrodynamic Gaussian wave packets over a wide range of amplitudes and spectral bandwidths. We show that when the envelope amplitude and width match those of the stationary NLSE soliton solution, the wave packet propagates steadily in the wave flume. More generally, by varying the amplitude of the Gaussian envelope, the wave packet may undergo either dispersive broadening or nonlinear focusing, resembling the dynamics of Satsuma-Yajima breathers. These findings may facilitate the controlled generation and manipulation of localized wave packets in a variety of nonlinear dispersive systems, including optics, plasmas, and Bose-Einstein condensates.

Being structures characterized by discrete rotational symmetry $\mathcal{C}_\nu$ of order $\nu$, photonic quasicrystals are capable to support stable propagation of linear vortex-carrying light beams and vortex solitons. However, the impact of discrete rotational symmetry $\nu$ of quasicrystals on the properties of vortex light states was not investigated so far, as only the systems constructed using the simplest Penrose tiling or corresponding optically induced Penrose lattices were considered in this context. Here we propose a broad class of quasicrystals with global topological defects – disclinations – introduced into their structure that allows to produce new quasicrystalline structures with any desired order of discrete rotational symmetry from basic Penrose structure. Such global topological deformation substantially enriches linear spectrum of quasicrystals, allowing them to support new types of linear vortex states and bifurcating from them families of stable thresholdless vortex solitons with unusual intensity and phase distributions. We found two different classes of stable vortex solitons consisting of in-phase or out-of-phase pairs of closely located bright spots, with total intensity distribution reflecting particular discrete rotational symmetry of the quasicrystal with disclination. Remarkably, even low-charge vortex solitons can be stable in quasicrystals with disclinations, while stability intervals for them broaden with decrease of the discrete rotational symmetry $\mathcal{C}_\nu$ of quasicrystal. Our results expand the theory of localization in quasicrystals to structures with global topological deformation, highlighting new prospects for robust transmission of power or information arising in these systems.

Symmetry reductions of partial differential equations (PDEs) inherit more geometric structures than other types of reductions: invariant conservation laws, variational structures, and, under suitable conditions, Hamiltonian-type structures of the original model descend to the reduced model through the mechanism of invariant reduction. This paper develops computational algorithms that carry out such reductions explicitly. We use the interpretation of variational $p$-forms as conservation laws of an enlarged system — the (degree-shifted) tangent system — consisting of the original equations together with their linearizations, in which the perturbation variables are treated as anticommuting. This allows us to formulate the algorithms using well-known concepts from the theory of conservation laws, naturally adapted to the graded-commutative setting. The reduction of conservation laws, variational $1$-forms, and presymplectic structures thereby becomes a single algorithmic procedure. We present (i) a homotopy-based reduction algorithm for systems of evolution equations, implemented in Maple; (ii) a descent reduction algorithm applicable to general $\ell$-normal systems for $p>0$; and (iii) a simple reduction algorithm available for point symmetries under suitable conditions, including the description of reductions in terms of systems involving fewer independent variables. Examples include nonlinear evolution equations in one and two spatial dimensions, the Laplace equation, the incompressible Euler equations, and the cotangent system of Pavlov's equation.

Non-KAM (Kolmogorov-Arnold-Moser) systems, when subjected to weak time-dependent perturbations, exhibit an abrupt transition to classical chaos through the breakdown of invariant phase-space tori. We showcase the utilization of non-KAM systems in the quantum regime as quantum sensors, leveraging their sensitivity at resonances. Quantum Fisher information (QFI) is a central quantity in quantum parameter estimation theory that measures how much information a quantum state contains about an unknown parameter that is encoded into it. In other words, it quantifies the sensitivity of a quantum state to small changes in that parameter. In this work, through numerical analysis in conjunction with analytical results, we study the performance of the non-KAM systems for quantum sensing applications by computing the QFI. We find that the growth of the QFI is remarkably enhanced when the resonance condition is satisfied. For frequency estimation under Floquet unitary encodings, we derive a transport bound: if the mean excitation number grows as $\langle\hat n(t)\rangle\sim t^{\alpha}$, the QFI obeys $I(t)\lesssim t^{2\alpha+2}$. The quantum kicked harmonic oscillator, a paradigmatic non-KAM system, realizes the full hierarchy: localized dynamics ($\alpha=0$) yield quadratic growth, delocalized diffusion along stochastic webs ($\alpha=1$) yields quartic growth, and translationally invariant resonances ($\alpha=2$) saturate the bound with anomalous hexic growth, $I(t)\sim t^{6}$, established analytically at resonance $R=2$ and numerically at $R=4$. The enhancement stems from resonance-induced translational symmetry rather than exponential instability, identifying non-KAM resonances as a metrological resource distinct from chaos-assisted and criticality-based sensing.

Self-heating couples the electrical and thermal states of filamentary memristors. However, the widely used Pickett compact model of $TiO_2$ resistive switching is isothermal and therefore cannot capture the resulting electrothermal dynamics. We introduce the Arrhenius-Thermal Filamentary Model (ATFM), which extends Pickett's tunneling-gap kinetics by incorporating a dynamic heat balance and an Arrhenius-activated switching rate. The resulting electrothermal feedback produces a sharp current-gated transition: below a critical drive current, the tunneling gap undergoes a non-returning ratchet drift, whereas above it, exponential locking of the filament kinetics establishes a bounded, drive-locked electrothermal oscillation. Using a stroboscopic Poincaré map and the Floquet multipliers of the resulting period-$1$ orbit, we characterize this onset as a threshold-like orbit contraction rather than a classical local bifurcation. In the limit $E_a\to0$, ATFM recovers the isothermal Pickett dynamics to numerical precision, as verified against an independent reference implementation over amplitude, frequency, and activation-energy sweeps. A variance-based Sobol' analysis with bootstrap confidence intervals identifies the excitation amplitude as the dominant control parameter and the thermal resistance $R_{th}$, rather than the thermal capacitance $C_{th}$, as the leading thermal contributor. A geometry-dependent temperature constraint further reveals a non-monotonic operating window in which an intermediate active area maximizes the switching excursion. The predicted trajectories are reproduced by both a fully behavioral SPICE netlist and a Verilog-A/OSDI device implementation, making ATFM directly suitable for circuit simulation. Overall, ATFM reveals and realizes a self-heating-driven dynamical regime within the widely used Pickett filamentary framework.

Extreme-mass-ratio inspirals (EMRIs) provide a unique laboratory for probing strong-field gravity and complex relativistic dynamics. We study a tidally deformed EMRI composed of a stellar-mass secondary orbiting a supermassive (non-)rotating black hole embedded in an external, adiabatically varying tidal environment. These systems provide a restricted, yet astrophysically motivated, realization of the relativistic three-body problem, expected to appear in active galactic nuclei, with a clear hierarchy of masses and radiation-reaction timescales. The external tidal deformation breaks the axisymmetry of the Kerr spacetime, rendering the geodesic dynamics non-integrable and giving rise to chaotic motion. The resulting signature of non-integrability is characterized through Poincaré maps and rotation curves constructed from the ratios of the fundamental frequencies of bound radial, polar, and azimuthal motion. We identify two prominent plateaus whose widths increase with the tidal field amplitude, signaling a transition from weak to strong chaos. We then demonstrate the sensitivity of the chaotic dynamics to the orientation of the orbit relative to the tidal field. We further analyze the proper-time evolution of the action-angle variables, showing that the angle combinations associated with the dominant commensurabilities are phase locked, thereby allowing the associated tidal contributions to induce secular changes in the constants of motion, whereas off-plateau angle combinations circulate. These results clarify the dynamical significance of the prominent plateaus and provide a novel phase-space characterization of tidal resonances in EMRIs. Finally, we discuss the potential role of radiation-reaction effects in driving EMRIs through tidal island crossings and the implications for gravitational-wave inference with future detectors.

Clusters that merge and split have no unique genealogy. Tracking the host size of a conserved constituent defines $q_*$ by $\langle(S_{t+\tau}/S_t)^{q_*}\rangle=1$, with $q_*\ne0$. Across coupled map networks, coagulation-fragmentation dynamics, and Vicsek systems, $q_*\simeq\alpha-2$ without fitting, where $c(s)\propto s^{-\alpha}$ on the same size window. Radius scans locate the known Vicsek interaction radius and recover an independently specified clustering scale in experimental active rods. Local scale covariance yields the relation, while a tent map counterexample shows when reducing the stationary balance at each parent size to one moment fails.

We construct isomonodromy transformations for linear $q$-difference systems of the form $Y(qz)=A(z)Y(z)$, where $A(z)=A_0+z A_1$ has diagonal leading coefficient. These transformations shift eigenvalues of the leading coefficient $A_1$ together with roots of $\det A(z)$. They lift compatibility to the right eigenpairs of $A(z)$, yielding a discrete local tau function. The resulting deformation equations preserve the Birkhoff connection matrix, and reduce in their $q\to 1$ limit to the isomonodromic deformation of a meromorphic connection on $\mathbb P^1$ with an irregular singularity of Poincaré rank one at $\infty$.

The exact N-soliton solutions are derived for the coupled negative first order Klein-Gordon (CNKG) equation subject to vanishing boundary conditions by using the inverse scattering method via Gelfand-Levitan-Marchenko equation. Based on the zero-curvature representation, the conservation laws, integrals of motion, and Hamiltonian structure of the aforementioned coupled nonlinear equations are constructed. The Jost functions and their analyticity properties for the Manakov spectral problem are recalled. The integral equations for the eigenfunctions are then used to formulate the Gel'fand-Levitan-Marchenko equations. Solving these equations establishes a direct correspondence between the kernel functions and the potential, yielding the general N-soliton expressions.

The Lorentz gas is a paradigmatic model in dynamical systems theory for understanding the origin of nonequilibrium transport in terms of microscopic deterministic chaos. In the periodic setting, a point particle scatters elastically off disks arranged on a two-dimensional lattice. Here we replace the disks by smooth potentials and the particle by the widely studied walking droplet, which propels itself on a vertically vibrating fluid. In the low-memory limit, this droplet reduces to a particle with nonlinear active friction. We call this system an active Lorentz gas. Using extensive numerical simulations, we analyze how dissipation generated by the active deterministic dynamics alters the phase-space structure of the corresponding conservative Lorentz gas. We find that islands of stability collapse into attracting and repelling sets. To characterize these structures, we introduce an energy-variance filtering method that distinguishes localized periodic, quasi-ballistic periodic, and chaotic trajectories, enabling the construction of bifurcation diagrams in a non-conservative setting. We identify parameter regimes exhibiting strong bifurcation cascades, anomalous diffusion, and significant phase-space contraction. Our results establish the active Lorentz gas as a rich framework for studying transport in dissipative dynamical systems and provide a bridge between active matter and classical models of chaotic transport, with potential implications for hydrodynamic quantum analogs in periodic media.

A central challenge in diagnosing quantum chaos is to distinguish genuine many-body scrambling from kinematic effects of the spectrum. Krylov state complexity, or spread complexity, has emerged as a powerful diagnostic, with its characteristic growth, peak, and relaxation often taken as signatures of chaos. However, previous work has shown that this criterion can give false positives: saddle-dominated integrable systems may display prominent peaks even without random-matrix level correlations. We argue, based on complementary numerical and analytical evidence, that this ambiguity can be resolved by unfolding the spectrum prior to constructing the ensuing Krylov dynamics. By removing the non-universal smooth density of states while retaining microscopic spectral correlations, unfolding suppresses spurious peaks in integrable systems while preserving the universal spectral signatures of chaotic systems. Analytically, the formulation of the Lanczos iteration in terms of orthogonal polynomials yields an exact complexity kernel with a robust near-diagonal structure whose fine-grained features reflect the underlying spectral correlations. Moreover, for the logarithmic model, unfolding can be performed exactly, mapping the spectrum to a uniform lattice and yielding an analytic spread complexity that removes the false-positive peak. These findings establish unfolded Krylov complexity as a more reliable probe of genuine many-body scrambling.

This chapter reconstructs the Hopfield network as a physical theory of memory rather than merely an early neural-network algorithm. It begins with the problem as it stood before 1982-threshold logic, Hebbian association, correlation memories, and recurrent binary networks-and isolates what Hopfield's synthesis added: a dynamical definition of content-addressable memory, a symmetric recurrent architecture with a Lyapunov function, a Hebbian embedding of patterns in its couplings, and a physical account of basins, robustness, and graceful degradation. The binary and graded-response energy functions are derived in full, together with the signal-crosstalk decomposition governing pattern stability, the mean-field theory of retrieval at extensive load, and the zero-temperature retrieval spinodal at (alpha 0.138) established by Amit, Gutfreund, and Sompolinsky. The energy-based program is then followed through analog optimization networks, polynomial dense associative memories, exponential interactions, and modern continuous Hopfield updates, including the precise conditions under which the update becomes scaled dot-product attention. Throughout, capacity claims are tied to their disorder ensemble, scaling limit, and success criterion, showing why numerically different storage limits need not conflict. A closing assessment distinguishes established results from surviving principles, assumption-bound limitations, and open problems, treating the Hopfield network as an effective theory whose symmetry, locality, and point-neuron assumptions delimit its biological reach. Fixed-seed numerical experiments expose the mechanisms discussed but do not substitute for analytical results.

We study how network connectivity and heterogeneous phase-delays shape the spatiotemporal dynamics of finite oscillator networks. Phase-delays can destabilize global synchronization and promote phase-locked patterns, including states with uniform phase gradients and more complex combinations of these modes. Yet, how connectivity and phase-delays jointly determine which states the network selects remains unclear. Here, we show that the spectrum of a composite matrix, which combines connectivity and phase-delays, governs not only the linear stability of the network's collective states but also their basin sizes. This, in turn, enables analytical estimates of basin size of phase-locked states for individual networks from connectivity and phase-delays alone. Applying this framework to nonlocal and global networks, including cases with random phase-delays, we uncover multistability and strong asymmetries in basin sizes, revealing chiral dynamics that conventional stability analysis cannot detect.

Reliable prediction of vehicle dynamics is essential for smart driving applications such as autonomous control and advanced driver-assistance systems. Off-road vehicles used in agricultural and construction settings are particularly prone to nonlinear behavior, including bifurcations and chaotic motion arising from intermittent loss of tire–road contact. Predicting such dynamics is challenging because it requires resolving both smooth nonlinearities and the discontinuous switching associated with contact loss. In this work, we investigate the feasibility of reservoir computing (RC) – specifically an echo state network (ESN) – for data-driven prediction of a jumping quarter-car model. The reservoir is trained on time-series data from a small number of points and evaluated on its ability to reconstruct bifurcation diagrams, phase-space attractors, and time trajectories across periodic and chaotic regimes. The trained reservoir qualitatively reproduces the period-doubling route to chaos, captures the geometric structure of periodic and chaotic attractors. These results demonstrate that reservoir computing is a feasible data-driven predictor of nonlinear dynamics in a practical, non-smooth vehicle system.

Reliable attractor recall conventionally requires broad basins of attraction. However, in reservoir-computing based associative memory, temporal cues reliably recover dynamical memories despite basins dominated by unpredictable, riddled-like regions. We reveal that memory basins exhibit an “octopus-like” structure: a robust “head” near the attractor and thin, intertwined “tentacles” spanning state space. Initial states in tentacular regions yield near-zero uncertainty exponents, making the recalled memory effectively unpredictable at finite precision. Yet, cue-driven generalized synchronization bypasses this unpredictability, driving the system into the robust basin head. This mechanism yields a quantitative relation linking minimum cue duration, synchronization rate, and basin-head radius. Trained recurrent neural networks exhibit similar geometry, suggesting this phenomenon extends beyond reservoir computing.

X-ray variability in compact-object binaries encodes the nonlinear dynamics of corona-jet interactions and accretion disk instabilities. Standard timing techniques characterize periodic and quasi-periodic variability well, but do not model underlying nonlinear dynamics or forecast their evolution. We apply Koopman operator theory and a data-driven approximation, extended dynamic mode decomposition (EDMD), to X-ray light curves for the first time. Koopman theory represents nonlinear evolution as an infinite-dimensional linear operator $\mathcal{K}$, whose eigendecomposition separates a complex system into independently evolving linear modes. We derive that each Koopman eigenfunction contributes a Lorentzian peak to the power spectrum, giving quasi-periodic oscillations a dynamical interpretation in which process noise damps modes and broadens their peaks. In both chaotic Duffing oscillator simulations and $\sim$30 yr of RXTE ASM and MAXI monitoring of the X-ray binary 4U 1705-44, the slowest-varying eigenfunction partitions state space into low- and high-flux regimes, changing sign days to weeks before transitions become visible in the light curve. Iterating $\mathcal{K}$ additionally yields flux forecasts on days-to-weeks horizons. The strengths of this framework are its generality across linear and nonlinear systems, its intrinsic interpretability through decomposed modes carrying explicit dynamical meaning, and its predictive power from propagating learned dynamics forward. These results establish Koopman operator theory as a new frontier of astrophysical timing and help advance interpretable machine learning for scientific discovery and understanding.

A population of $N$ identical pulse-coupled oscillators ultimately settles into one of two outcomes: full synchrony or a state of co-existing synchronized clusters. We show that which outcome occurs is controlled by the prime factorization of $N$. At the critical charging curve — linear, the boundary between the synchronizing and clustering regimes — the synchronization basin acquires exact arithmetic structure. The synchronization probability is $\Psync=A_{N,1}/N^N$ for all $N$, where $A_{N,1}$ satisfies an exact recurrence relation. For prime $N$, $A_{N,1}=N^N-1$ giving the closed form $\Psync=1-1/N^N$; for composite $N$, the observed asymptotic scaling is $1-\Psync\sim C_m N^{-(m-1)}$, where $m$ is the smallest prime divisor. The result adds a new member to the atlas of exotic basin geometries: alongside fractal, riddled, and tentacled basins, we now have a basin that is arithmetic.

An exact zero of a generated harmonic needs two real cancellations; a damped oscillator's leading fundamental-source balance supplies one. Absorber loss leaves a phase deficit of fixed sign, amplitude independent at leading order, closed instead by the orbit's own $5\Omega$ motion returning to $3\Omega$ at weight $|\Delta_1|^4$. Closure is therefore unreachable at weak amplitude: the channel supplying it ranks at $O(F^7)$, not at the leading $O(F^3)$. The fifth harmonic is $1.8\times10^{-3}$ of the relative-coordinate fundamental, yet a balance homotopy weighting it below $0.48$ annihilates the tracked pair in a saddle-node.

The chaotic dynamics of a large one-dimensional lattice of nonlinear maps with a spatial coupling of increasing extent is explored using the covariant Lyapunov vectors (CLVs). A theoretical approach is developed that predicts the spectral variation of the CLVs and the spectrum of Lyapunov exponents. The theoretical approach is independent of the details of the mapping function and requires only a description of the lattice coupling. The general structure of the spatial power spectrum of the CLVs is determined using only the eigenvectors of the coupling matrix. This yields a decomposition of the CLVs into two regimes. The first regime, corresponding to the largest Lyapunov exponents, contains highly entangled CLVs with significant contributions from large scale spatial structures and aligns with the sorted eigenvectors, which occur in pairs with wavenumbers that increase linearly with their index. The second regime contains significant mixing of length scales with increasing index. The index separating these two regimes is approximately equal to the fractal dimension of the dynamics.

Continuous cellular automata such as Lenia spontaneously produce lifelike, self-propelling patterns, including the Orbium glider, which travels in a straight line while pulsing through a fixed breathing cycle. Collision-based logic, where moving patterns compute by colliding, is established in discrete cellular automata and continuous physical media. Within continuous cellular automata, computation has so far been trained into the rule rather than emerging from collisions, and whether a fixed-rule automaton like Lenia can support general collision-based computation remains open. This paper constructs an INHIBIT gate from collisions of the Orbium glider. Of the patterns searched across four continuous-CA rule types, the Orbium glider is the only one shown to survive a collision with both copies intact. A control glider deflects a signal glider off its output line, so the output carries a signal only when no control is present. The gate blocks across all twenty-four phases of the breathing cycle and nine integer offsets of the control. Two such gates in series, with one signal line and two controls, compose into an AND-NOT chain, correct on all eight input combinations. By contrast, routing a signal beyond that single chain is undemonstrated. A deflected signal is not restored to a fixed landing position, and no reusable absorber for the surviving gliders was found. The immediate open question for collision-based computation in Lenia therefore narrows from whether a gate exists to whether a deflected signal can be delivered to a downstream gate, the next requirement for composing the gate beyond a single straight chain.

We study linear instabilities and pattern formation in a spatially extended Selkov model for glycolysis with diffusion and chemotactic interactions between two species. Linear stability analysis reveals two distinct bifurcations: a zero-wavevector, finite-frequency Hopf bifurcation leading to spatially homogeneous oscillations, and a finite-wavevector Turing instability with a saddle-node bifurcation leading to stationary inhomogeneous patterns. The selected wavevector, $k_c$, depends sensitively on the nature and strength of chemotaxis and varies nonmonotonically with the chemotaxis parameters, allowing re-entrant transitions between homogeneous and patterned states. For sufficiently strong chemotaxis, when the two species mutually attract or mutually repel each other, an unusual instability emerges in which the preferred wavevector diverges as a finite threshold in chemotactic strength is approached. We complement the linear analysis with extensive direct numerical simulations (DNS) of the nonlinear partial differential equations in two dimensions. The DNS confirm the predicted instabilities and their nonmonotonic and re-entrant character, and also reveal spatially uniform oscillatory states for suitable parameters. The stationary patterns include spots and, unexpectedly, stable stripes. We further uncover transitions between spot and stripe morphologies controlled by chemotactic strength and diffusivity ratios. The stability of the striped states is consistent with linear amplitude equations, whereas the spot-stripe transition is attributed to nonlinear effects beyond linear stability. Our results demonstrate that chemotaxis can act as a nonmonotonic control parameter for instability selection, wavelength selection, re-entrant pattern formation, and nonlinear pattern morphology in reaction-diffusion systems.

We address the existence and dynamics of solitons propagating along helical (spiral-rotating) and straight trajectories in the bulk waveguide with the cubic-quintic nonlinearity and a single- or multi-ring potential. The input is taken as a vortex beam with a single or multiple phase singularities (pivots), displaced from the waveguide's axis. The single off-axis pivot creates a one-ring helical soliton, with a closed or open (split) ring structure. The vortex beams with multiple off-center pivots give rise to complex states with multi-ring, multi-core, and/or multi-split structures. The multi-core/split solitons propagate along helical channels, or straight ones, which are parallel to the propagation axis. The sign and period of the helicity can be precisely adjusted by applying a torque with an appropriate angular velocity.

Coupled logistic maps exhibit collective ordering of their directional phases. As the system parameter varies, the directional phases can undergo a transition from an in-phase state to an anti-phase state, while the individual map trajectories remain chaotic. In this work, we propose a data-driven machine learning (ML) framework based on parameter-aware reservoir computing (PARC) to predict order-parameter dynamics in two representative systems: a logistic map and a two-dimensional coupled map lattice (CML). For the logistic map, the reservoir is trained using only pre-crisis time series data at bifurcation parameter $\mu$ values below the attractor-merging crisis ($\mu_0 = 3.6786$). The trained reservoir reconstructs the full bifurcation diagram and correctly predicts the transition in the directional order parameter $M(\mu)$, from an ordered state ($M \approx 0$) to a disordered state ($M \neq 0$) across the crisis point. For the CML, we exploit the spatial homogeneity of the lattice: a single reservoir is trained on the dynamics of one representative lattice site and is then replicated across all $L^2$ sites during prediction, where $L$=50. The replicated reservoir correctly predicts the transition from in-phase synchronization ($\theta \approx 1$) to anti-phase clustered states ($\theta \approx 0$) at $\mu \approx 3.82$, where $\theta$ quantifies phase coherence across lattice sites.

Cellular automata conform a set of computational models which evolve with a reduced set of simple rules, yet still are able to show extremely complex emergent phenomena such as fractals and universal computation. Despite their apparent simplicity, they have shown great potential in simulating natural systems and solving challenging computational tasks such as classification and image generation. Instead of implementing cellular automata purely at the software level, it is desirable to design novel analog computing platforms that physically evolve following the automata's underlying rules, thereby reducing power requirements and latency. Here, we introduce an optical extreme learning machine for the simulation of a wide range of cellular automata. Our system operates in free space, and uses a spatial light modulator to encode the evolution rules of the system, while coherent wave propagation performs the corresponding computations. Our results demonstrate a simple, fully-programmable, cost and power efficient, and easy to build and align platform for the implementation of a wide range of complex computational systems such as elementary cellular automata, Conway's Game of Life, and two-dimensional Turing machines.

Ensuring a sustainable energy supply requires maintaining power-grid stability. Rotor dynamics are governed by the swing equation, which takes the form of a second-order Kuramoto model with a correlation between power and total coupling strength. Yet the microscopic mechanisms that nucleate and propagate synchronized clusters remain poorly understood. Using a minimal model motivated by empirical grid data and the \Adhoc potential method, we reveal two distinct seed cluster types and propagation pathways: a population-driven seed cluster at the center of the power distribution propagating to its tails by rotor accretion, and, for a symmetric distribution, coupling-driven seed clusters at its tails propagating inward to the center by cluster merger. These differences generate distinct order-parameter patterns, while inertia controls whether the seed clusters persist with distinct angular velocities. The same propagation pathways also govern recovery following external disturbances. We further confirm that the same selection–persistence rule holds in data-derived annealed representations of European power grids. Therefore, our results can inform strategies for sustaining stable power-grid operation. More generally, our pathway-based framework reframes synchronization by emphasizing the dynamics of cluster formation and recovery rather than relying solely on static criteria.

Many natural and engineered systems are large complex networks of interacting components, and external perturbations drive them along different dynamical paths. Identifying which perturbation occurred matters for diagnosis, control, and prediction. Yet often times only a few components can be jointly monitored. Which components should be monitored? Experimental practice usually favors placing reporters at the most sensitive sites, where perturbations produce the largest effects. Using a simple dynamical model for complex systems with heterogeneous connectivity, we ask how sparse reporter panels should be chosen to classify shocks from partial trajectories when repeated trials only approximately reproduce an ideal initial condition. Once that reproduction is imperfect, sensitivity ranked panels fall far short of optimal, and the shortfall grows with the noise. We find that the best panels mix two kinds of reporters. A promiscuous reporter responds to most shocks, so it separates them mainly by degree, and degree fluctuates from trial to trial. A dormant reporter responds to only a few shocks but strongly, and its answers do not scatter as much between trials. As noise grows, the cost of losing a dormant reporter rises to meet the cost of losing a promiscuous one. Panels of either kind alone classify worse than the mixture, and no property of the members collected individually explains the ordering. Most of all, we find that only a minuscule number of reporters are needed on a panel to accurately identify which shock hit the system. We implement an efficient algorithm to assemble such a panel. Together these results provide a low cost practical design principle for monitoring large complex dynamical systems.

The classical problem of a rigid body in an ideal fluid is a six-dimensional dynamical system with three conserved quantities. Despite its long history and its importance as a reduced model of many fluid-structure problems, basic questions remain, including the shapes and connectivities of the three-dimensional submanifolds of solutions, and the nature of chaotic motions on these. We explore the problem with a particular ellipsoidal body, beginning with integrable motions— steady linear translation and rotation and periodic planar tumbling and fluttering. The stability of these states depends on the relative magnitudes of linear and angular momentum or energy. Linear stability, including Floquet, analysis of lower-dimensional perturbed systems is consistent with direct integration of the full system. We document instabilities leading to a variety of regular and chaotic motions, including flipping and twirling, whose trajectories appear to follow paths near connections between the integrable states. Such indirect observation of these connections provides insight into the structure underlying the rich dynamics of this simple system. For the submanifold where linear and angular momentum are perpendicular, we further examine how solutions fit together to fill momentum space in different zones of stability of the integrable states, and thereby uncover additional bifurcations in connectivity and what appear to be new integrable solutions.