Papers

We propose a unified perspective on classical and quantum chaos based on the sensitivity of a system's stationary states to slow driving. We probe this sensitivity via the system's susceptibility to the average protocol speed, which we call the “speed-Fisher information," and relate it to irreversible entropy production in the system. We show that chaotic dynamics manifests as a divergence of the speed-Fisher information with the protocol time, and that this response is controlled by the perturbation's low-frequency spectral weight. This approach to chaos applies to both classical and quantum Hamiltonian systems, and naturally extends to non-Hamiltonian classical flows. Additionally, we identify a quantum regime at times shorter than the Planckian timescale, in which the speed-Fisher information exhibits a uniquely quantum scaling. We illustrate this framework with simple classical and quantum examples

Time series prediction is a common application of reservoir computing. When the training and testing time series data contains multiple dynamical regimes, because an underlying parameter is changing, or the data in fact consists of multiple distinct systems, simple application of the reservoir computing principle produces high prediction errors. Here, we propose a HyperReservoir as an extended model of reservoir computing especially designed for such cases. The HyperReservoir combines a main reservoir with a smaller context reservoir, where the latter modulates the output weights of the former. This structure resembles the hypernetworks from deep neural network literature. However, in contrast, HyperReservoirs retain the simple training via linear regression of standard reservoir computing. We compare the proposed architecture with a conventional ESN, in which context acts at the input, and a full-matrix Conceptor, in which context modulates the reservoir state space. We evaluate all three models on time-series prediction tasks based on Lorenz and Rössler systems, including for varying bifurcation parameters and time sampling scales. We find that the HyperReservoir achieves the lowest mean test error in all three tasks, and particularly outperforms conceptors on data that is sampled from the same attractor but at different time scales.

Digital quantum simulation (DQS) is one of the most promising applications of quantum computing. Trotterised DQS models quantum dynamics by decomposing the evolution into a sequence of gate-based Trotter steps. Using larger Trotter steps reduces the required gate count, but it is now understood that the Trotterisation step size cannot be increased beyond a certain threshold value, beyond which there is a sudden breakdown in simulation accuracy, due to the onset of quantum chaos. To date, this behaviour has only been studied for closed systems not subjected to the environmental noise and decoherence to which real-world quantum computers are highly sensitive. In this work, we study a range of experimentally relevant open-system quantum simulator models and show that Trotterisation thresholds persist even when open-system effects are included, even in the long-time, steady-state limit, once all coherence has been lost. We show that open DQS can accurately reproduce the dynamics of open target systems prior to the threshold, but still exhibits a sudden performance breakdown beyond it. We introduce new, statistical techniques to rigorously characterise the onset of quantum chaos beyond the Trotterisation threshold for both approximately unitary and genuine dissipative quantum chaos, based on random matrix theory (RMT), building on techniques developed for closed systems. Finally, we demonstrate that dissipative quantum chaos emerges when the dissipation is in balance with the presence of underlying unitary quantum chaos; the dissipation must be strong enough to induce genuinely dissipative behaviour, yet not so strong as to suppress the underlying chaotic structure.

In many turbulent flows, the quantities of interest are coarse-grained variables, either because resolving the fine scales is computationally prohibitive or because only coarse observations are available; in both cases, data-driven reduced-order models that evolve only these variables are required. Even when the governing equations are known, their projection onto the coarse variables is not closed, and the central difficulty lies in representing the feedback of the unresolved scales so that the reduced model reproduces the statistical and dynamical properties of the projected high-resolution system. We address this problem with a score-based framework, constructing a reduced-order model of quasi-geostrophic turbulence whose stationary distribution coincides by construction with the distribution learned from data and whose dynamics are calibrated to reproduce the finite-time correlations of the energies in individual wavenumber shells. The resulting model reproduces the stationary statistics of the resolved flow with high accuracy, and its calibration is computationally inexpensive because it requires no forward integration of the model.

Reduced descriptions of unresolved fluctuations are commonly based on Gaussian processes, although many realistic forcings have finite correlation times and a renewal structure. For memoryless step (Kubo–Anderson) noise, we show that the reduced dynamics admit two exact and complementary descriptions: a kernel representation governed by the Boolean cumulants of the jump distribution, and a frozen-noise representation adapted to stationary probability densities. For exponentially distributed waiting times, the totally time-ordered $G$-cumulants coincide with the Boolean cumulants of the jump law, and the memory kernel is resummed exactly as the Boolean generator $\eta$ evaluated on a resolvent operator. Second-order closure is exact if and only if the jump law is symmetric Bernoulli; otherwise, the leading closure error is controlled by $(b_4/b_2)\lambda^2$. The Boolean hierarchy, however, acts on the kernel, not on the stationary measure. Stationary densities are approximated by replacing the jump law with its $N$-point Gauss quadrature: each surrogate preserves all multi-time correlations up to order $2N-1$ for any waiting-time law, its kernel is a Padé resummation of the Boolean one, and it is itself an exactly solvable renewal problem. For the linear system with linear multiplicative interaction (LIMI/CAM), the frozen-noise representation reduces the dynamics to a random affine recursion and yields exact support boundaries, singularity exponents and Kesten tail indices, confirmed numerically. Rates, moments and closure errors are governed by the Boolean hierarchy, whereas stationary densities are determined by the geometry of the jump distribution. For memoryless renewal noise, the combinatorial structure controlling finite-correlation reductions is Boolean.

The long-run statistics of chaotic dissipative partial differential equations are described by invariant measures, and when the equation carries a parameter these measures form a family whose members can differ in kind, from steady states to sustained spatiotemporal chaos. We ask whether a single generative model can cover such a family. We train a single conditional diffusion model once, on states of the Kuramoto-Sivashinsky equation at 160 values of its hyperviscosity, and evaluate it against held-out data at interleaved parameter values not seen in training, using a classifier two-sample test supported by spectral, geometric and tail statistics. Over most of the family, agreement improves as the dynamics become more chaotic. We identify a structural cause: the weakly chaotic members have invariant measures concentrated on low-dimensional sets, which a model with full support output cannot represent, and the target approaches full dimensionality as chaos increases. Restricting training to the strongly chaotic members reduces the classifier's excess over chance on those members by roughly 40%, and this restricted conditional model matches or exceeds models trained at single parameter values. On those members, generated samples reproduce energy spectra to within a resampling floor, with a small systematic deficit at the largest scales, and match the energy balance of held-out data.

Even if it were possible, attempting to model the motion of a fluid in a pipe by tracking each individual molecule would be an extraordinary waste of computation. The emergent laws of fluid mechanics accomplish the same goal on a laptop instead of a datacenter. But a sufficiently complex fluid such as in Earth's atmosphere again requires a datacenter, this time to resolve the many contortions in the fields making up our weather. Might there be some analogous set of emergent laws that would represent the effect of these perturbations without needing their explicit representation on a numerical grid? We argue that the existence of a symmetry principle implies such laws are likely to exist. The atmosphere is scale invariant, implying that the observed structure at one scale is nothing but a stretched version of that at any other scale, at least within the invariant regime. Though some emergent laws built on scale invariance exist, they are built on a challenging ontological foundation, making them hard to develop further. We propose a path forward, first by introducing a new fundamental building block of the atmosphere that we term a “turbulon”, inspired by a lesser-known generalization of turbulence theory that accounts for buoyancy. We then construct the Superposition of Turbulons and Eddies Atmospheric Model (STEAM), and show that simulated atmospheric volumes are broadly plausible when compared to state-of-the-art hydrodynamic simulation output. Some aspects of STEAM clearly need improvement, though other statistics reproduce observations better than hydrodynamic models, and visualizations of simulated clouds are strikingly realistic. We estimate STEAM's computational cost to be up to a million times less than hydrodynamic models, suggesting that the Navier-Stokes equations may not, in fact, be required for simulation and prediction of Earth's weather and climate.

The Yukawa-Sachdev-Ye-Kitaev (YSYK) model, a solvable model of non-Fermi liquids with a proposed cavity-QED realization, couples $N$ complex fermions to $M$ bosons of frequency $\omega_0$ through random Yukawa couplings of strength $g$, and its dynamics depends only on $R\equiv\omega_0/g^{2/3}$. For $R\ll1$ the bosons are slow and act as a static random background in which the fermions are free, while for $R\gg1$ they are only virtually excited and generate a random four-fermion interaction of rank $M$, which is generically chaotic for $M\geq2$. Using exact diagonalization at infinite temperature, we follow the emergence of random-matrix statistics between these limits with unfolded spectral statistics and the Krylov complexity of the thermofield-double state. For complex couplings the level-spacing distribution evolves from Poisson to Gaussian-unitary-ensemble (GUE) statistics, the spectral form factor develops a correlation hole and a ramp, and the Krylov complexity evaluated on the unfolded spectrum develops a peak above its late-time plateau whose height reaches the GUE value. All diagnostics locate the onset around $R\approx0.3$, and adding boson modes moves it to smaller $R$. This delineates the parameter regime in which a cavity-QED realization would display random-matrix spectral correlations. On the other hand, real couplings give the same crossover towards Gaussian-orthogonal-ensemble (GOE) statistics, and along an interpolation between the two the peak height follows the mean gap ratio, so the peak identifies the symmetry class as well as the onset of chaos. Without unfolding the peak stays about 20$\%$ below its random-matrix value in both classes, which we trace to the non-semicircular density of states, and at larger $R$ the boson-number bands slow down the spreading of the state in Krylov space.

Modern deep neural networks treat parameters as static floating-point matrices stored in physical memory, incurring Von Neumann memory bottlenecks and representation collapse. We formulate Orbital Error Dynamics (OED), an analytical framework wherein synaptic weights are not stored masses (O(W)), but transient topological resonances (O(1)) derived procedurally from the complex quadratic polynomial map z_{n+1} = z_n^2 + c. We introduce the Bent Sine Wave Hypothesis, demonstrating that non-equilibrium living systems emerge when harmonic waves curl inward through environmental drag toward the cardioid cusp (c = 1/4). We define the Observer Horizon Geometry in parameter space, identifying interior resonance shoulder loci X_upper = (0.25, +0.18) and X_lower = (0.25, -0.18) between the fixed-point basin and the true boundary at c = 0.25 +/- 0.50i. To escape non-convex stagnation without loss zeroing, we introduce a heavy-tailed Biomimetic Perturbed Jump Operator (Omega_tunneling) inspired by mammalian fertilization zinc sparks. We further couple an enteric-cranial Dual-Brain architecture shielded by adaptive CD4+ regulatory immune gating (M_CD4), and project the 4-nucleotide genetic basis (A, T, C, G) across quadrants in C. Multi-seed empirical validation on the Two-Moons manifold (5 seeds, 80/20 train/test split, 32x32 grid, zero test-time updates, zero label leakage) demonstrates that procedural parameterization from a 24-byte coordinate seed achieves 77.67% +/- 5.35% clean test accuracy (within an 8.00-point paired difference of an unconstrained gradient baseline at 85.67% +/- 5.35%, 95% CI: [-1.07%, 17.07%]) and 71.33% +/- 3.80% under distribution shift (N(1.2, 0.4)), alongside conceptual equivalence with an analog optical co-processor.

Integer maps are discrete dynamical systems defined on the natural numbers. In this paper, we investigate the dynamics of the shifted Alladi-Erd{\H o}s map, a one-parameter family of integer maps in which each composite number is mapped to the sum of its prime factors, while each prime is mapped to $n+A$, where $A \in \mathbb{N}$ is a fixed shift parameter. By systematically exploring the parameter space for $2 \leq A \leq 10^5$, we uncover a rich bifurcation structure characterized by the emergence, disappearance, and reorganization of attractor cycles as the shift parameter varies. We find that although several attractors may coexist for a given value of $A$, for most values of $A$, almost all natural numbers belong to the basins of only two dominant attractors. Using elementary number theoretic arguments, we explain the observed bifurcation diagram. We further characterize the attractor cycles and quantify the distribution of their basin sizes across the parameter space. Our results reveal an unexpectedly rich landscape of arithmetic dynamics arising from a remarkably simple integer map. %and provide a systematic characterization of how attractors and their basins evolve as shift parameter is varied.

Chimera states, characterized by the coexistence of coherent and incoherent dynamics in networks of coupled oscillators, are among the most intriguing collective phenomena in nonlinear systems and are strongly shaped by the underlying topology and initial conditions. Their analysis in large-scale networks remains computationally demanding and motivates the development of coarse-graining strategies that reduce network size while retaining the essential dynamical features of chimera behavior. In this Letter, we investigate whether equitable-partition-based quotient graphs can preserve chimera patterns under substantial reductions of network size. Using nonlocally coupled FitzHugh-Nagumo (FHN) oscillators, we exploit the fact that, for an equitable partition, the full-network dynamics restricted to the associated cluster-synchronous subspace coincide exactly with the quotient dynamics. We ask whether this structural exactness is sufficient to preserve chimera signatures after the network size reduction. Our results show that this is not necessarily the case: agreement between quotient and full-network chimera dynamics depends on the equitable partition, and exact quotient dynamics do not necessarily guarantee the transverse stability of the corresponding full-network trajectory. Importantly, when the cluster-synchronous solution becomes transversely unstable, the full network may depart from the exact quotient trajectory while remaining in a macroscopic chimera regime. Therefore, structural exactness and dynamical preservation are distinct requirements, and transverse stability provides a complementary criterion for assessing quotient-based coarse-graining of chimera states.

In the pursuit of steady-state fusion energy, Advanced Tokamak regimes rely on Internal Transport Barriers (ITBs) with reversed magnetic shear. While Resonant Magnetic Perturbations (RMPs) are commonly applied to control Edge Localized Modes (ELMs) by intentionally making the plasma edge chaotic, the RMPs can also affect the reversed-shear cores. To address this compatibility, we propose a new, adjustable analytical model for plasma current that explicitly incorporates the core, ITB, and edge pedestal components. By applying this current formulation to a Hamiltonian map, we compare how easily magnetic chaos reaches the core in ITB-driven reversed-shear (Non-Twist) plasmas. We find that the localized ITB current creates "twin" inner resonances and pushes outer magnetic island chains closer to the edge. These results highlight a critical operational trade-off: while the Non-Twist topology makes ELM control easier at lower coil currents, it severely shrinks the safety margin against a complete, RMP-induced loss of global confinement.

Two-dimensional Kolmogorov flow is a common proving ground for the dynamical systems picture of turbulence. Hundreds of unstable periodic orbits have been found for this system across many Reynolds numbers and domain sizes, although the most well explored is Re$=40$ on a square $2\pi\times 2\pi$ domain with a forcing wavenumber of $k=4.$ Here I present a catalog of 8952 exact steady states of the flow in this canonical parameter regime. These solutions provide a fixed constellation informing the dynamics of both turbulence and transients.

Many physical data assimilation (DA) workflows require smoothing methods that represent non-Gaussian posteriors over physical state variables, scale to high-dimensional simulators, train from observation windows alone, and remain compatible with calibration of the prescribed simulator. We introduce PR-Smoother, a simulator-preserving amortized smoother designed for this prescribed-simulator DA regime. Its key design principle is to keep the prescribed simulator explicit in both the evidence lower bound and the variational family: rather than learning replacement dynamics or a learned trajectory prior, PR-Smoother learns only future-conditioned corrections around the prescribed rollout. This yields an explicit non-Gaussian smoothing distribution over physical trajectories and supports joint state, parameter, and sensor-bias learning from observations alone. The variational family contains the exact smoother in deterministic and linear-Gaussian limits. Empirically, PR-Smoother captures multimodal posteriors in 4-dimensional Lorenz-96, remains accurate under ambiguous nonlinear observations and process noise in 40-dimensional Lorenz-96, and scales to joint state-parameter-bias inference in 16,384-dimensional Kolmogorov flow.

We present an analytical organizing principle underlying the dynamics of the periodically forced Duffing oscillator. In the massless limit the system reduces to a time-dependent cubic equation whose discriminant and Jacobian classify the canonical Duffing regimes and determine the number and stability of instantaneous equilibria. For the double-well oscillator, the same construction gives an exact forcing threshold $F_{\mathrm{bif}}$ separating intra-well and inter-well dynamics. We show that this algebraic quantity governs the finite-mass dynamics far beyond the singular limit from which it originates: the boundary between intra-well and inter-well motion converges to $F_{\mathrm{bif}}$ as $m\to0$, and the families of pitchfork bifurcations responsible for asymmetric inter-well states accumulate at a common point. Numerical simulations further show how the discontinuous jumps of the degenerate model are regularized at finite mass into short oscillatory transients that shrink as $m\to0$. These results reveal the massless Duffing equation as the organizing center of the full finite-mass dynamics and establish a direct link between the algebraic structure of the degenerate problem and the global bifurcation structure of the oscillator.

This paper investigates dynamical properties of the Nonlinear Walecka Model (NLWM), also known as Quantum Hadrodynamics (QHD), which provides a relativistic framework for describing nuclear matter through nucleon-meson interactions. While the original linear model successfully captures qualitative features of nuclear matter, it overestimates nuclear compressibility; consequently, nonlinear extensions incorporating cubic and quartic self-interaction terms in the scalar field $(\sigma)$ are employed to achieve better agreement with experimental data. By treating the self consistent effective-mass equation as an iterative mapping, the study explores the emergence of complex behaviors such as periodic orbits and chaotic regimes. Using tools from nonlinear dynamics, including return maps and bifurcation diagrams, we characterize routes to chaos via period doubling cascades and identify the presence of "shrimps" stable isoperiodic islands within chaotic regions of the parameter space. Furthermore, we discusses the physical implications of these findings, suggesting that crisis phenomena and structural convergence may serve as dynamical signatures of macroscopic rearrangements in the scalar mean field, potentially influencing the stability and equation of state (EOS) of dense nuclear matter in extreme environments like neutron stars.

Mass-conserving reaction-diffusion systems with two species correspond to a seemingly simple case where pattern formation occurs under the influence of a conservation law. Here, we first revisit their linear stability behavior and point out that generically two instabilities can occur: a stationary large-scale mass-conserving (Cahn-Hilliard) instability and an instability that combines features of a stationary large-scale non-mass-conserving (Allen-Cahn) instability and a oscillatory large-scale mass-conserving (conserved-Hopf) instability. We term it an Allen-Cahn-Hopf instability. Second, we investigate the nonlinear dynamics for a specific model related to the formation of cell polarization where only a Cahn-Hilliard instability can occur, i.e., all primary bifurcations are stationary. We analyze how secondary and further bifurcations subsequently give rise to various oscillatory states. The emerging rich spectrum of spatiotemporal behavior includes several period-doubling cascades related to different forms of spatial and temporal symmetry breaking. Beside regular states, three types of low-dimensional spatiotemporal chaos occur and involve transitions like fusion and an outer crises. Our results demonstrate the importance of nonlinear interactions in the dynamics of mass-conserving reaction-diffusion systems, and show that even a simple two-species system with primary bifurcations of Cahn-Hilliard type can show complex spatiotemporal behavior.

The trajectories of point vortices after a division using reverse collapse route are studied. In this setting, an unexpected peculiar phenomenon appears, corresponding to the shadowing of the vortex breakup. This occurs at least for a finite time, when the system evolves from three to five vortices after a breakup. An analytical study reveals that this observed numerical phenomenon is related to the condition of scale invariance, a condition necessary for the finite time singularity. Further simulations agree with our analytical findings and indicate that shadowing can occur with more than five vortices, and as well with higher order singularities. This phenomenon leads to question what a vortex is actually measuring, triggering speculations if this shadowing property could be generalized to other Hamiltonian systems with finite-time singularities.

Context: The eccentric von Zeipel-Lidov-Kozai effect, which is widely applied to diverse astrophysical settings, can drive the inner binary to extremely high eccentricities, where short-range forces such as general relativity (GR) become prominent. Aims: Poincare surfaces of section show that GR effects reshape phase-space structures, giving rise to a widespread and large-scale chaotic sea. This work aims to uncover the underlying mechanism of GR-enabled chaos. Methods: The dynamical structures are studied within an adiabatic framework, where an adiabatic invariant is constructed. Phase portraits, defined by the level curves of this invariant, establish analytical boundaries for the chaotic domains. Numerical simulations are subsequently performed to map orbital flips, maximum eccentricities, and Fast Lyapunov Indicators (FLI) across the initial eccentricity-inclination parameter space. Results: Phase portrait analysis demonstrates that GR precession introduces an uncertainty zone near polar inclinations, where trajectories crossing this zone inherently become chaotic. Leveraging this framework, the chaotic boundaries across the full parameter space are analytically derived, yielding excellent agreement with numerical maps of flipping orbits, maximum eccentricity, and FLI. Conclusions: In the presence of short-range forces, flipping orbits accompanied by extreme eccentricity excitation are fundamentally chaotic, and the underlying mechanism of large-scale chaos originates from the periodic passage of trajectories through the uncertainty zone in phase space over secular timescales.

Intermittent dynamics are ubiquitous in the Earth system and often arise from the interaction of processes evolving on different time scales. In this work, we investigate how intermittent bursts in a fast forcing system propagate to and reshape the dynamics of a slower response system that would otherwise settle onto a quasi-stationary or weakly oscillatory regime. We address this question in two coupled models of increasing complexity: a low-dimensional Lorenz-63 system and the spatially extended Kuramoto-Sivashinsky equation. Across both systems, intermittency in the forcing progressively reshapes the attractor of the slow response system and drives it into different regimes. Using the Wasserstein distance, we show that increasing the frequency of intermittent events progressively displaces the response attractor from its unperturbed counterpart, up to a limit beyond which this deviation saturates. We then show that varying the forcing intensity and the time-scale separation between the forcing and response systems drives distinct regime transitions, which we characterize through the variance of ensemble maxima, the power spectra of both systems, and extreme value statistics. Finally, we examine how the type of intermittency in the forcing system affects synchronization between the two systems through local phase locking, showing that specific transitions in the phase-locking behavior are tied to the underlying intermittency regime, and that the response delay scales exponentially with the time-scale separation.

We investigate next generation reservoir computing (NGRC) as a data-driven approach for inferring unseen components of dynamical systems. We compare NGRC with traditional reservoir computing (RC) using the Lorenz and Rössler system, where two unknown components are inferred from one given component. For both systems, NGRC achieves accurate results while requiring fewer training data and less computational time than RC. We identified an inverse proportional behavior between the number of time-delayed steps needed for NGRC and the temporal resolution, indicating that the physical time span covered by the delay interval is an important factor in determining the required number of delayed steps. Finally, we apply NGRC to the observational climate data of ENSO (El Niño–Southern Oscillation) and infer one observable from the remaining variables. Despite the noise and complexity of the real-world data, the NGRC shows promising results. Our findings demonstrate the potential of NGRC for efficient inference of unseen components in both controlled dynamical systems and real-world data.

The logistic map is a quintessential model in the study of low-dimensional chaos. High-dimensional chaos, on the other hand, presents itself in disordered systems with many interacting and heterogeneously coupled components. Here, we formulate a system of many logistic maps, interacting through disordered couplings. Using a combination of dynamic mean-field theory, random matrix theory and numerical simulations, we show that even the smallest amount of disorder can remove the period-doubling cascade in the conventional logistic map. Instead we find a transition to high-dimensional chaos, marked by an oscillatory instability not previously reported for disordered systems. We also show that with sufficiently strong homogeneous coupling between the maps one recovers elements of the conventional period-doubling cascade. Our findings indicate that well-known phenomena in dynamical systems can be fragile in the face of disorder. At the same time, new phenomena emerge that are neither found in simple low-dimensional dynamics nor in high-dimensional disordered systems.

We resolve an existing question concerning dynamical and structural stability of the E {\times} B staircases in magnetically confined fusion plasma. The problem has stirred considerable interest in the literature due to a unique combination of its scientific and practical aspects. Here, we report a new approach to the problem by combining the paradigmatic Hasegawa-Wakatani model of electrostatic drift-wave turbulence with an idea of interaction renormalization, imported from quantum field theory. It is shown based on the Schrödinger-Newton equation with renormalized interaction constant that the plasma staircase is robust and stable as soon as the nonadiabaticity parameter, {\delta}, goes above a certain critical value, and is unstable otherwise. This critical value is an exact result of the interaction model and is found to be {\delta}c = {\pi}. For {\delta} < {\delta}c , the unstable system expands in the radial direction, resulting in a radial second moment that grows with time t as t^(2/3). The results obtained bear important implications with regard to the implementation of arrays of semipermeable transport barriers in tokamaks, suggesting a threshold condition for their generation.

The hypersensitivity and turnpike phenomena in the optimal control of an inverted pendulum are investigated from a dynamical-systems perspective. We show that, for a fixed terminal time and a fixed terminal state optimal control problem, (1) the hypersensitivity originates from the fractal structure of the set of initial adjoint variables in the associated Hamiltonian dynamics, (2) the turnpike arises from slow dynamics in the vicinity of a degenerate center manifold, and (3) the escape channels are formed by normally hyperbolic invariant manifolds (NHIMs). As a consequence, small perturbations in the initial adjoint variables lead to qualitatively distinct extremal trajectories, resulting in severe numerical instability in trajectory optimization. Both the fractal structure and the invariant sets are characterized numerically and analytically.

Equilibria of the Hamiltonian system associated with Pontryagin's minimum principle exhibit an exact doubling of the natural spectrum and, under a simple pairing condition, a Jordan block at every simple purely imaginary eigenvalue. Consequently, the classical Lyapunov Center Theorem does not apply to the augmented system, and no periodic orbit with nonzero optimal control bifurcates at linear order. We establish this mechanism in general and show that an optimal-control-induced periodic family emerges at cubic order in a Lindstedt–Poincaré expansion. The mechanism is illustrated in closed form for the pendulum and evaluated numerically for the planar $L_2$ equilibrium of Hill's restricted three-body problem, where the third-order approximation is validated against an independently computed family of periodic orbits.

We present an equation-free method for maintaining a dynamical system operating in a desired regime, even as system parameters are disturbed in such a way that qualitatively different dynamics would emerge. This method allows for real-time identification of changes made to a system's evolution equations induced by the parameter changes. If the equations governing the original system's dynamics are known, our method can allow one to determine the values of the parameters in real time. We illustrate our method with numerical examples using a reservoir computer control implementation.

Estimating the dynamic aperture of circular particle accelerators typically requires expensive long-term tracking. Using classical homology, we construct a two-dimensional theoretical classification of regular and chaotic orbits and use persistent homology for its numerical implementation on short orbit segments. In four dimensions, we introduce topological chaos indicators based on persistent homology and use normal-form theory to apply the classification to two symplectic-plane projections. The methods reproduce the expected organization of regular and chaotic motion in several canonical Hamiltonian systems. Applied to symplectified one-turn maps of a nonlinear lattice model of the Integrable Optics Test Accelerator, the results obtained from the first 200 turns identify the central connected stability region and its surrounding transition to chaos. They are consistent with the short-term dynamic aperture of the electron configuration obtained from $10^4$ turns and the long-term dynamic aperture of the proton configuration obtained from $10^8$ turns. For the configurations studied, persistent homology of short orbit segments predicts the dynamic aperture obtained by substantially longer tracking, potentially reducing the need for exhaustive simulations in studies of existing accelerators and the design of future facilities. The methods could also complement classical chaos indicators and modern approaches based on artificial intelligence.

The Lorenz system is the canonical low-order model of convective instability, yet its dissipative and driving terms have never been checked against, nor constructed from, an explicit thermodynamic bookkeeping. We derive a modification that satisfies the local-detailed-balance condition for macroscopic relaxation toward nonequilibrium steady states, thereby identifying the thermodynamic force, entropy-production rate and frenesy of the resulting flow. The resulting model produces a transition from a quiescent fixed point to a robust, large-amplitude relaxation oscillation, closely analogous to recharge-discharge oscillator paradigms used for the El Nino-Southern Oscillation. The system alternates between a long, nearly reversible recharge phase and a brief, violently frenetic discharge burst, during which essentially all of the cycle's activity and entropy production is concentrated.

Multiple pendulum with branching is proposed as a convenient platform to study energy transfer between different modes. The antiphase oscillation mode is localized to the "child" links, which makes it easy to prepare initial conditions without exciting the antiphase mode. A manageable expression for the energy transfer is derived theoretically and evaluated with experimental data.

We study the off-diagonal matrix elements of real-space observables in time-reversal-invariant single-particle chaotic systems. By analyzing the semiclassical expression for the off-diagonal variance derived from Berry's conjecture, we show that the banded structure of the observable matrix emerges naturally. For local observables, we identify a characteristic bandwidth associated with a late-time timescale inversely proportional to the particle velocity. We further show that, for systems with steep-wall confinement, the predicted magnitude follows the entropy scaling of the eigenstate thermalization hypothesis (ETH), multiplied by an additional kinetic-energy-dependent factor that is independent of spatial dimension and is not captured by conventional many-body ETH. We illustrate these results through a case study of quantum billiards and verify the semiclassical scaling numerically in a generalized quarter-Sinai billiard. Our results elucidate the dynamical implications of Berry's conjecture and provide a comparison between single-particle eigenstate thermalization and many-body ETH.