Papers

One of the central challenges in evolutionary dynamics is understanding why some species combinations persist while others disappear. Although cyclic-interaction models have provided fundamental insights into biodiversity maintenance, much less is known about how hierarchical competitive interactions shape long-term community organization. Here, we investigate a hierarchical extension of the May-Leonard model, in which species interact through a directed predation chain while undergoing reproduction and mortality. Combining mean-field analysis with Monte Carlo simulations, we show that the fully coexisting state is generically unstable, causing the dynamics to evolve toward lower-dimensional coexistence states. The simulations further reveal stochastic extinctions dominating small populations with the dynamics progressively approaching the mean-field predictions as the system size increases. Rather than permitting arbitrary species combinations, the hierarchical-interaction structure dynamically constrains coexistence by selecting only specific subsets of species for long-term persistence. We show that these admissible coexistence states have a natural graph-theoretic interpretation as independent sets in the hierarchical interaction network, thereby providing general constraints on coexistence in hierarchical communities. Together, these results establish a theoretical framework linking hierarchical interactions, dynamical selection, graph topology, and biodiversity organization, extending the classical May-Leonard model beyond cyclic competition.

Can asymmetry between two interacting networks fundamentally change how they synchronize? We identify double explosive transitions in the forward direction, backward direction, or a combination thereof, with single or double hysteresis loops in an adaptive bilayer multiplex network of Kuramoto oscillators with pairwise and three-body interactions and asymmetric phase lags. Using the Ott-Antonsen reduction, we derive a low-dimensional system and perform a stability analysis. The reduced model accurately captures the full microscopic dynamics and enables analytical expressions for the bifurcation points. Systematic mapping across multiple parameter planes reveals eight distinct synchronization regimes. The relative ordering of saddle-node and pitchfork bifurcation points—controlled by phase-lag asymmetry, cross-layer adaptation, and the higher-order interaction strength—creates two distinct coherent branches (weak and strong), giving rise to double explosive transitions. Crucially, phase-lag asymmetry acts as a robust control knob: while symmetric phase lags suppress explosive transitions, layer-specific differences promote multistability and double explosive transitions. The higher-order interaction strength $K_2$ and adaptation strengths $q$, $p$, and $h$ further modulate these transitions in a complex, parameter-dependent manner. Excellent agreement between analytical predictions and numerical simulations confirms the robustness of our reduced description.

Artificial feedback spin masers exhibit rich nonlinear dynamics, yet a systematic framework for understanding and controlling their behaviors remains elusive. In this paper, we consider the polarity and geometric configuration of the feedback field as important control parameters for studying the nonlinear dynamics of spin masers. By cooperatively tuning the strength and phase delay of feedback field, we investigate the spin dynamics at various configurations of the feedback field. Periodic, quasiperiodic and chaotic spin oscillations are uncovered. Specifically, a unique spin oscillation mode akin to the time crystal effects emerges within a certain parameter regime. This work presents a different viewpoint understanding the nonlinear dynamics of spin masers including the time crystal.

This thesis investigates the impact of resonant mode-particle interactions on transport and confinement in toroidal fusion plasmas. Using analytical and numerical approaches, we study how intrinsic and externally applied magnetic perturbations affect plasma particles with different kinetic characteristics, focusing on resonant interactions with guiding-center motion. We develop a novel, computationally efficient method based on Action-Angle variables to identify resonance locations and characterize resonances in guiding-center phase space, including the number of islands in resonance chains and the formation of transport barriers. Using the drift center (DC) approximation, we derive analytical expressions for orbital frequencies and the kinetic $q$ factor for large-aspect-ratio (LAR) equilibria. These results provide the conditions for mode-particle resonances, which can strongly influence particle, momentum, and energy transport and, consequently, plasma confinement. The analytical results are validated against numerical simulations, demonstrating an efficient tool for predicting transport barriers and energetic-particle behavior under non-axisymmetric perturbations. We further extend the orbital-frequency analysis from LAR equilibria to numerically reconstructed, realistic equilibria using a computationally efficient, semi-analytical geometrical method applicable to arbitrary unperturbed equilibria. Finally, for the LAR equilibrium, we extend the analysis to time-dependent perturbations and demonstrate the emergence of Arnold diffusion in guiding-center phase space, highlighting the role of the Arnold web in particle transport. Overall, this work advances the understanding of mode-particle resonant interactions and provides computational tools for predicting transport and confinement in realistic fusion configurations.

The Julia set is defined by the closure of repelling periodic points in chaotic systems. Why does this structure not appear in integrable systems? In this paper, we address this question by demonstrating the existence of the closure of divergences of the periodic equations, which we designate as the "anti-Julia set." We also call the sets before taking the closures the pre-Julia and pre-anti-Julia sets, respectively. We illustrate the transition mechanism by considering a complex map that interpolates between integrable and non-integrable dynamics, by introducing a real deformation parameter $a$. For $0<a\le 1/2$, we show that the Julia set and the anti-Julia set coincide in the complex plane, although the pre-Julia and pre-anti-Julia sets remain completely disjoint. At the integrable limit $a\to 0$, these two dual structures undergo a critical collision and subsequent annihilation, reminiscent of a cosmological Big Crunch. When $1/2<a<1$, on the other hand, the boundary of the pre-anti-Julia set includes the Julia set. We analytically characterize these phenomena, focusing in particular on the asymptotic behavior of the pre-anti-Julia set as it approaches the integrable limit, and provide numerical visualizations that elucidate the underlying mechanisms of this Big Crunch phenomenon.

Infectious diseases may interact by competing for the same hosts or by facilitating subsequent infections. Understanding the dynamics of such multi-pathogen systems, particularly those subject to endemic seasonality and mitigation, is essential for designing robust public health interventions. We propose a three-stage modeling framework to disentangle the interplay between seasonality and behavioral feedback as a way of mitigation. First, we analyze a coupled susceptible-infectious-recovered-susceptible (SIRS) system without external forcing and show that the abrupt transition between the disease-free and endemic equilibria arises from a backward bifurcation-induced first-order phase transition. Second, we independently examine seasonality and behavioral feedback, characterizing where and when oscillatory behavior is induced near critical tipping points. Third, we demonstrate that their combination generates complex multi-annual wave patterns, with high-incidence cycles driven by seasonality and low-incidence intervals driven by behavioral feedback. By mapping stability as a function of seasonal forcing, mitigation strength, and cooperativity, we identify distinct period-doubling cascades with chaotic signatures arising from different mechanisms: the interplay between seasonality and behavior, and inter-pathogen cooperativity with the backward bifurcation it induces. We then analyze how these mechanisms interact across parameter ranges. Altogether, we show that cooperation fundamentally expands the spectrum of possible epidemic patterns, highlighting the importance of considering multi-pathogen interactions in epidemic modeling and control strategies.

A Poincaré section replaces a flow by a return map, but for a chaotic system this map is usually known only from sampled crossings. We show that coarse transport can be read directly from Koopman spectral data, without fitting the map. Measure-preserving EDMD retains the isometric structure; riggedDMD then approximates spectral measures and constructs finite regularized wave packets. Packet phase supplies a finite-resolution transport coordinate; low modulus marks a singular skeleton where the phase becomes ill-conditioned. We demonstrate the idea on the Rössler system, a 32-mode Kuramoto–Sivashinsky Galerkin system, and the forced Duffing oscillator. The packets yield coarse symbolic models on sections ranging from an almost one-dimensional curve to a visibly thick set. Their graphs organize observed low-period orbits and guide targeted searches for others. In Duffing Regime~II, a seven-region rule accounts for $91\%$–$94\%$ of filtered one-step transitions, while failures in the lowest retained modulus decile occur at $5.08$–$5.20$ times the overall rate. The packets are not Koopman eigenfunctions, nor are the regions exact Markov partitions. Together these computations show how spectral information beyond isolated eigenpairs can expose chaotic transport directly from trajectories.

Physical reservoir computing (PRC) with neuromorphic networks offers a promising approach to brain-inspired information processing, exploiting emergent nonlinear dynamics of physical neural networks as a computational resource. This study demonstrates fully autonomous closed-loop prediction of the Mackey–Glass (MG) chaotic time series using a simulated neuromorphic nanowire network as the physical reservoir. Two strategies are evaluated: the virtual node (VN) method, which expands the feature space by temporal multiplexing of reservoir states, and a non-VN approach that uses all physical node readouts directly without temporal multiplexing. Results are reported for two values of the MG time delay parameter, $\tau = 18$ and $\tau = 21$, the latter representing a more complex chaotic regime not previously evaluated for this class of physical reservoir. Over a short prediction horizon of $T = 100$ timesteps, the VN approach achieves autonomous prediction accuracies of $90.4$% and $89.7$% at $\tau = 18$ and $\tau = 21$, respectively, while the non-VN approach achieves $81.5$% and $76.2$%. Long-horizon analysis over $T = 500$ timesteps shows that both approaches reproduce the qualitative attractor structure and dominant spectral content of the true MG signal, with trajectories remaining bounded throughout. These results suggest that the intrinsic dynamics of neuromorphic nanowire networks are sufficient to support meaningful autonomous chaotic time series prediction without virtual node augmentation, and that performance may improve further as physical network sizes scale to the millions of nodes achievable in hardware. As this study uses simulated networks, extrapolation to physically fabricated large-scale arrays remains to be validated experimentally.

Framewise variance normalization removes fluctuation amplitude from an observation, but not from the dynamics that generate later patterns. The current normalized spectrum therefore need not determine its own future. We study this effect in three two-dimensional phase-separation models that change transport, add active driving, or suppress large-scale separation. For paired initial fields, their normalized spectra evolve identically in the linear regime. The restoring model provides an exact example: damping does not directly change how the normalized spectrum evolves when both equations are applied to the same field. It nevertheless changes the hidden amplitude, which alters the later nonlinear shape. An exact comparison separates the immediate effect of changing the equation from the effect of reaching a different state through earlier evolution. It also reveals strong cancellation between these effects in the mobility and active models. As a practical test, spectral histories improve classification by about 4-6 percentage points at several later observation windows, although the gain depends on quench depth and spectral construction. Temporal spectra can therefore reveal dynamical consequences of information absent from every individual normalized frame.

We present a Hessian-based approach to numerically continue periodic orbits in dynamical systems. A loop (periodic orbit candidate) is parametrized as a Fourier series; a loss function is defined based on the deviation of the loop from the physical differential equations. Unlike previous work relying on hand-derived Jacobians, our method automates the process by leveraging automatic differentiation, a common machine learning technique. The continuation direction can be determined by the flat directions of the loss landscapes (directions with zero eigenvalues), making the search of periodic orbits efficient and guided. Our method is integrator-free, precisely initializes oscillations around unstable fixed points, and efficiently detects orbit family intersections and subharmonic bifurcations. As a demonstration, we present full continuations of periodic double pendulum oscillations from fixed points, showing bifurcations along orbit families and categorizing branches of periodic orbits. In particular, we find periodic orbits where both pendulum masses are never simultaneously at rest, which to our knowledge has been missing in the literature.

Chaotic evolution measures how rapidly nearby configurations separate, but the existence of such rapid evolution does not by itself determine what physical information the evolution actually carries. For fixed-coupling JIMWLK evolution with fresh transverse-local, color-isotropic noise, we show that this distinction becomes exact - leading to a no go theorem. At finite rapidity step and finite transverse regulator, the exact linearized update depends on the Wilson-line background only through local adjoint rotations of fresh color-isotropic noise. Once the past is conditioned upon, these rotations leave the complete local noise measure invariant, so the tangent process has the same probability law for any target ensemble evolved with the same regulated fixed-coupling dynamics, including dilute and saturated ensembles. After the exact global color-rotation mode is removed, the reduced tangent process found to have a positive leading Lyapunov exponent which indicate chaotic evolution. Fixed-coupling JIMWLK therefor can be dynamically chaotic while its tangent-only statistics carry no information about the target ensemble. Spatially nonlocal noise however evades the argument because correlations between distinct sites leave relative Wilson lines in the conditional covariance.

We present a simple algebraic proof of meromorphic non-integrability of a 'mysterious' case of a quadratic Hamiltonian studied by Sokolov and Wolf. The result is obtained by applying the Morales-Ramis method, based on differential Galois theory.

Detecting weak transient signals in noise is a persistent challenge in sensing, communication, and electrophysiology. Here, we demonstrate a physical weak-signal detector based on a semiconductor negative differential resistance (NDR) device operated near a Hopf bifurcation. A coherent input that persists over the response time of the dynamical system can drive a transition between quiescent and oscillatory states, whereas faster stochastic fluctuations are largely suppressed. This nonlinear transformation converts a weak analog threshold crossing into all-or-none voltage spikes and therefore provides asynchronous signal detection without a reference clock. Using a modulated photovoltaic signal, we detect a weak frequency component of 100 Hz as a demonstration, at an input signal-to-noise amplitude ratio as low as 1/500, and reliably recover it in the output spectrum. We then apply the same principle to neuronal multisite extracellular recordings. After standard band-pass filtering, the raw microelectrode signal is transformed by the NDR dynamics, enhancing the distinction between neuronal spikes and background fluctuations. The resulting detected spike times agree closely with those obtained using a traditional spike-detection pipeline. These results establish bifurcation-engineered NDR dynamics as a compact hardware approach to noise-resilient signal discrimination and event-based analog-to-digital conversion that will be useful for neuroprosthetic devices.

Unstable periodic orbits and steady states are believed to form the backbone of spatiotemporal chaos and turbulence, yet their computation in thermally driven flows remains scarce for transitional regimes. In this work we compute and characterize a steady state and three families of periodic orbits in two-dimensional Rayleigh-Bénard at $\mathrm{Pr}=1$, near the transition to chaos. We find that in its route to chaos, the flow hops between several sets of orbits after becoming quasiperiodic. We use Floquet analysis to study the stability of the orbits obtained and characterize their bifurcations, showing how the appearance of primary and secondary frequencies, as well as phase-locking mechanisms, are all related to the dynamics of the orbits. We study in detail how the flow shadows the orbits found and determine in which regimes each orbit is dynamically relevant or not. Our analysis also reveals two important insights: (1) all symmetries are broken before the onset of chaos, and (2) this onset does not alter the behavior of the heat transport.

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.

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.

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.

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.

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.

We reformulate the relaxation problem in statistical mechanics by making explicit what are the operational objects subject to relaxation: the local time statistics of the recorded signal $Z(t)$. These local time statistics are simply the estimated histograms of observations $\{Z(t_i)\}_{i=1}^M$ performed at uniformly random times $\{t_i\}_{i=1}^M$ by a clockless observer. The subject of prediction is a belief about a future fresh out-of-sample reading of a measurement outcome whose distribution is inferred from the mathematical model believed to be true. For finite bounded systems of $N\ge 1$ degrees of freedom global irreversible relaxation of predictions can occur but special initial conditions exist. The form of the predictions depends on certain loss functions whose choice is up to the particular observer. Finally, entropy is given a learning interpretation as mutual information between the observer and the unknown past of the system under consideration and, in complete generality, its stationary value depends on the information available.

We investigate the statistical properties of the non-hierarchical planar three-body problem, relevant to astrophysical systems with nearly planar dynamics, including protoplanetary disks and AGN disks. Using the Flux-based statistical theory of the three-body problem, previously studied in three dimensions, we derive theoretical predictions for several statistical observables in the planar case. Focusing on ergodic disintegration outcomes, we compare these predictions directly with numerical simulations and contrast the results with the corresponding unconstrained three-dimensional case. Our analysis is based on one million simulations for each of eight distinct mass sets. We examine escape probabilities, properties of marginal escape events, eccentricity distributions, lifetime distributions, and relative prevalence of prograde and retrograde escapes. While several of the theoretical predictions are in good agreement with the simulations, others exhibit unexpected discrepancies. We discuss possible explanations for these deviations.