Papers

We introduce a trigonometric version of the Nosé-Hoover oscillator in which the quadratic mechanical terms and unbounded thermostat coupling are replaced by bounded trigonometric functions. This formulation replaces the harmonic potential by a pendulum-type potential and confines the thermostat interaction to a bounded periodic form. The resulting two-parameter system is naturally defined on the three-dimensional torus and reduces near the origin, to leading order, to the classical polynomial Nosé-Hoover model. We investigate its global dynamics using Poincaré sections, bifurcation diagrams, Lyapunov spectra, Kaplan-Yorke dimensions, and the Lyapunov Integrability Test (LIT). The numerical results reveal the coexistence of regular and chaotic dynamics and characterize changes in dissipative behavior across the parameter plane. We then analyze the two limiting cases associated with the parameter axes. For $a=0$, we construct two functionally independent first integrals on regular domains, whereas for $b=0$ the dynamics reduces to a family of two-dimensional systems on invariant tori, which are analyzed using Darboux polynomials and exponential factors. First-order averaging near the intersection of these integrable limits yields periodic solutions bifurcating from unperturbed periodic orbits and an obstruction to regular $C^1$ first integrals in their neighborhoods. Independently, differential Galois theory applied to the normal variational equation, together with the Ayoul-Zung and Li-Shi criteria, excludes meromorphic $B$-integrability and non-constant meromorphic first integrals near a particular non-equilibrium phase curve for $ab\neq0$. Thus, despite retaining the local structure of the classical Nosé-Hoover oscillator, its trigonometric counterpart exhibits markedly different global dynamics and integrability.

It is often assumed that general stationary solutions to the wave kinetic equation are described solely by conserved fluxes and temperature. The families of solutions parameterized by these quantities are referred to as warm cascades, and are derivable from differential models of wave kinetics. These models rely on the assumption of local interactions, and we demonstrate by a time scale argument that the nonlocal effect of thermal damping on the cascade may result in dependence on an additional parameter, the UV cutoff, calling into question the validity of warm cascade solutions. This is shown for systems with both three-wave and four-wave interactions. We then consider the depletion of the cascade by this damping, and present a numerical demonstration of this nonlocal and nonuniversal behavior.

What can and cannot be predicted by a model when only limited information is available? We answer this question by constructing a ladder of information limits that organizes prediction tasks according to their information requirements: from memoryless and memory-augmented trajectory forecasts to event prediction, stationary statistics, and generative laws. The analysis, which is independent of the model structure or architecture, identifies the minimum attainable error along with the information limitations that give rise to it. The approach distinguishes information hidden in unresolved variables from information recovered through time-delayed observations and connects these contributions to Mori–Zwanzig memory. The ladder also reveals why Lyapunov growth alone cannot characterize reduced-order prediction error, why stationary statistics may remain predictable even when individual trajectories become unpredictable, and why stochasticity can represent (but cannot recover) missing information. Numerical studies of the Kuramoto–Sivashinsky equation and the Lorenz system illustrate these results across the ladder. This unified perspective provides a principled language for understanding the fundamental limitations of ROMs for a given prediction task and for clarifying whether improved predictions require richer observations, greater precision in the input, or additional memory.

Learning in recurrent neural networks can fundamentally reshape their underlying dynamics, transforming initially chaotic activity into stable task-dependent behavior. We develop a non-equilibrium dynamical mean-field theory(DMFT) to describe this transition during learning. We show that a slow feedback-driven learning process generates an evolving effective feedback strength that drives the network through a transition from chaotic to stable dynamics defined by a bifurcation of the DMFT solution. By deriving the two-time correlation function throughout learning, we identify a critical feedback strength and a corresponding learning rate dependent critical time separating these regimes. The transition arises from the progressive deformation of an effective dynamical landscape by the growing learned feedback structure. Starting from the untrained state, the theory predicts the time evolution of the network output during training and shows quantitative agreement with numerical simulations.

Chaos emerges when stretching is repeatedly recycled by reinjection. We uncover a new route to chaos in which the decisive variable is the temporal order of non-normal tangent maps: periodic and chaotic states can share essentially the same one-step stretching statistics while their ordered products acquire opposite Lyapunov growth. We introduce the ordered-product growth rate $h_L$ over $L$ successive tangent maps, which reveals how states indistinguishable at one step separate under geometric composition and identifies the finite composition scale at which chaos emerges. We use this mechanism to establish a new form of global chaos control: minute phase actions reorient the successive non-normal amplification directions so that their geometric composition becomes contracting, suppressing chaos at fixed dissipation without reducing local amplification or targeting a preselected orbit.

We study the Arnold-Beltrami-Childress (ABC) flow in perturbative regimes near its three elementary integrable coordinate axes. Starting from a degenerate family of periodic streamlines of the limiting integrable system, we use first-order averaging to prove the existence of two isolated periodic solutions of the perturbed flow. The three perturbative regimes are related by the cyclic symmetry of the ABC vector field. In each case, the corresponding two-dimensional averaged system possesses two simple equilibria, one elliptic and the other hyperbolic, depending on the sign of the relevant parameter ratio. Poincaré sections illustrate how the degenerate periodic structure of the integrable limit breaks under perturbation and how the isolated periodic orbits predicted by averaging emerge within the surrounding dynamics. These periodic orbits provide the natural link between the perturbative analysis and the integrability problem. The eigenvalues of the linearized averaged system determine the leading-order behavior of the nontrivial characteristic multipliers of the bifurcating periodic solutions. Combining this relation with the Poincaré-Llibre-Valls criterion, we prove that, in a neighborhood of each of these periodic orbits, there exists no nonconstant first integral $H\in C^1$ that is regular along the orbit. The analytical results are further illustrated and confirmed by direct shooting, monodromy-matrix computations, and continuation of the elliptic and hyperbolic branches, demonstrating that the first-order averaging approximation remains quantitatively accurate over a substantially wider parameter range than the strict asymptotic regime.

In nonlinear prediction, two state representations with the same local uncertainty volume can have radically different predictive value. We consider differentiable finite-time dynamics together with a quadratic terminal requirement that specifies which terminal state differences matter. Pulling this requirement back through the tangent map defines a requirement-induced predictive geometry on the present tangent space. This geometry is fixed by the dynamics and the requirement; under the fixed local uncertainty-volume constraint used here, the capacity parameter fixes only the scale of the representation that realizes it. We show that the unique optimal covariance has the inverse predictive shape and, equivalently, that optimality is characterized by isotropization of propagated uncertainty in the terminal requirement geometry. The gain over a baseline-isotropic representation is the arithmetic-to-geometric mean ratio of the predictive spectrum, making it a coordinate-invariant measure of predictive anisotropy. In Euclidean coordinates this yields a general finite-time spectral law, and for two-dimensional area-preserving dynamics an exact identity between predictive gain and the condition number of the optimal geometry. Stronger directional stretching therefore increases potential predictive gain only together with the anisotropy required to realize it; bounding that anisotropy imposes a finite gain ceiling. Standard-map and double-pendulum calculations demonstrate these relations in nonlinear Hamiltonian systems. The framework provides a basis for adaptive sensing, state estimation, and model-predictive computation in which representation geometry is matched to predicted task consequences.

Repeated decisions exhibit temporal structures such as persistence, direction-dependent switching, recurrent alternation, and abrupt transitions. We examine the generative sufficiency of a two-dimensional fast-slow dynamical system. The system combines a cubic fast equation with linear slow feedback and is analyzed through its equilibrium geometry, trace-determinant structure, equilibrium-fold loci, candidate Hopf boundaries, and singular critical manifold. An explicit observation operator projects continuous trajectories to a scalar signal and applies a binary readout, separating latent state-space dynamics from observable behavior. The analysis establishes a unique-equilibrium regime and, for suitable parameters, a three-equilibrium wedge with a central saddle. The outer equilibria are attracting only where their traces are negative. The analysis also identifies the simple-zero condition required for ordinary saddle-nodes, trace-zero positive-determinant spectral boundaries compatible with oscillatory instability, and the attracting and repelling branches of the critical manifold. Prescribed nonautonomous sweeps numerically illustrate direction-dependent switching (T1), transient episodic recurrent switching (T2), and an abrupt localized regime shift (T3). The attracting-equilibrium regime associated with prolonged state retention (T4) is characterized analytically. The resulting correspondence is intended as a test of generative sufficiency at the level of observable temporal organization, rather than as an identification or empirical validation of latent cognitive mechanisms.

We present a direct and nonrecursive construction that maps an arbitrary planar polynomial dissipative flow $\dot{\alpha}=h(\alpha,\alpha^*)$ to an open quantum system in Gorini–Kossakowski–Sudarshan–Lindblad form. Given the homogeneous components of the classical drift, the corresponding Hamiltonian and collapse-operator blocks are obtained algebraically and independently at each degree. No recursive cancellation of lower-order terms is required: in the large-amplitude limit $|\alpha|\sim S\to\infty$, ordering-generated lower-degree terms are parametrically suppressed, while mean-field closure yields the prescribed leading $O(S)$ drift for semiclassically localized states. This provides a simple and systematic route from dissipative classical dynamics to explicit open-quantum-system realizations, without claiming a unique microscopic quantization of the classical flow. We demonstrate the construction for stable fixed points, a Hopf bifurcation, and a bistable-ring flow. The corresponding Liouvillian spectra recover the classical relaxation exponents, the radial Floquet exponent and neutral phase direction of a limit cycle, and the separation between local relaxation and inter-ring switching in the bistable case. These examples show that the construction can be used not only to analyze a given quantum model, but also to design open quantum systems with prescribed classical dynamical structures in the semiclassical limit.

Hamiltonian and Lagrangian mechanics are powerful frameworks for analyzing physical systems. Previous work has extended these formalisms to ecological systems, such as the predator-prey Lotka-Volterra (LV) system. In this Article, we derive a canonical Lagrangian for the two-dimensional LV model directly from its Hamiltonian representation. We find that the two-dimensional LV system admits a standard canonical Lagrangian formulation with one degree of freedom and a non-quadratic kinetic structure. This formulation admits a mechanical interpretation in terms of a particle moving in a potential well, where the non-standard kinetic structure produces a position-dependent damping term that can instead act as "revving." The derivation provides a direct connection between predator-prey dynamics and a canonical formulation of mechanical dynamics. As a verification of the construction, we apply Noether's procedure to the explicitly time-independent derived Lagrangian and reveal that the well-known Hamiltonian of the LV system is the corresponding conserved quantity. We also uncover a subtle redundancy associated with the choice of canonical momentum and its identification with the original population variables.

A chemical transition state is a phase-space bottleneck: trajectories approach along stable directions and leave along unstable directions. Lagrangian betweenness (LB), introduced in finite-time transport theory for fluid flows, combines backward and forward deformation in a way suggestive of this gather-and-disperse geometry. We ask what LB measures when the transition-state geometry is known. For a linear rank-one saddle, LB is spatially constant even though the stable and unstable manifolds are present. Relative nonlinear corrections remain small on shrinking initial neighborhoods, including observation times on the logarithmic local escape scale. At fixed spatial resolution, however, increasing observation time can concentrate the spatial variation of LB near a stable or unstable manifold. A separable quartic Hamiltonian makes these two limits explicit. A nonseparable Hamiltonian shows local flattening near a hyperbolic periodic orbit. We then use HCN/CNH isomerization, where dividing surfaces and gap times are established by phase-space transition-state theory. Before any exit, larger LB generally accompanies longer eventual gap times. The incoming reactive action provides a dynamical interpretation: initial conditions closer to the normally hyperbolic invariant manifold have longer local passages and greater accumulated stretching during passage. Among trajectories still inside at 0.5 ps that exit before 5 ps, early LB does not usefully rank exit times. LB therefore characterizes transition-state-organized transport in this example; a large value is neither an intrinsic manifold marker nor a general measure of molecular residence time.

We observe the emergence of extreme events in an active fluid involving two distinct chemical species that regulate active stress. One species is slow diffusing and the other is fast diffusing, and the growth of the fast-diffusing species is modelled using a nonlinear logistic term. We demonstrate the presence of extreme events in the temporal evolution of the concentrations, as well as in the spatial profile of the system,in regimes of merging-emerging soliton-like dynamics and spatio-temporal chaos, through analysis of the time-series, bifurcation diagrams, probability distribution functions of the concentration of the two species, return maps and distributions of inter-event intervals. Interestingly, we also find evidence of pronounced bunching of extreme events and super-extreme events in the slow chemical species in the soliton-like regime. We go on to systematically explore the dependence of the extreme event occurrences on the Péclet number and the strength of the nonlinear growth term, and find that the probability of extreme events increases after a critical Péclet number, while increasing the nonlinearity suppresses extreme events. Lastly, in order to gain further insight, we investigate a modified mode-truncated reduced order model comprising of coupled differential equations mimicking this active fluid system. We find that this reduced order model also exhibits extreme events whose emergence is correlated with a sudden expansion in attractor size due to a crisis arising from attractor collision.So these results demonstrate the existence of extreme events in an active fluid system, and are of potential relevance to biological phenomena where active transport plays an important role.

Classical theories of random neural networks typically assume independent connectivity, overlooking the local motif structures prevalent in biological circuits. Here, we investigate how four second-order synaptic motifs (chain, reciprocal, convergent, and divergent) shape the dynamics of nonlinear firing-rate networks. While previous studies have established that chain correlations generate outlier eigenvalues, we demonstrate that these motifs also jointly reshape the Jacobian eigenvalue bulk. Using the path-integral formalism, we derive a dynamic mean-field theory which reveals that the chain motif acts as a retarded feedback of the ensemble-mean activity through the response kernel, producing a rich repertoire of dynamical regimes, including ferromagnetic states and limit cycles. At sufficiently large magnitude, negative chain correlations produce a glassy, multistable regime that was previously mainly associated with partially symmetric networks. Our theory also distinguishes convergent from divergent motifs: divergent correlations primarily rescale temporal noise, while convergent correlations suppress temporal chaos by converting nonzero mean activity into quenched heterogeneity. Finally, analyses of the Lyapunov spectrum and participation-ratio dimension show that motif structure changes the geometry of chaotic activity, reducing entropy production and attractor dimensionality even when the effective spectral edge is held fixed. Together, these findings establish second-order motifs as a fundamental structural mechanism governing the dynamical regimes of local cortical circuits.

We examine a 4-dimensional generalization of predator-prey models describing interactions among four species. We show that these systems exhibit a rich variety of asymptotic behaviors, with solutions that are either integrable, leading to non-sustainable dynamics characterized by unbounded population growth or collapse, or chaotic, where all species coexist over time. Moreover, we review the integrable cases, which are classified into two-parametric families, and examine three-parametric families that behave similarly to the known two-parametric integrable cases. We also analyze chaotic Lotka-Volterra models in terms of their Lyapunov exponents and Poincare' surfaces of section, which turn out to reveal underlying structures that are compatible with what is expected from sustainable dynamics.

We investigate a discontinuous route from integrability to chaos using a confined stochastic random walk and a deterministic stadium-like billiard. In both systems, the stationary diffusive observable exhibits a finite jump at the transition: it vanishes at the unperturbed limit but approaches a finite, geometry-controlled value for arbitrarily small nonzero perturbations, providing the characteristic order-parameter signature of a first-order transition. Despite this discontinuity, the relaxation timescale diverges as the transition is approached, revealing critical slowing down. Both models exhibit normal diffusion with $\beta=1/2$, a perturbation-independent stationary state with $\alpha=0$, and a crossover iteration scaling as $n_x\propto\lambda^{-2}$, yielding $z=-2$, where $\lambda$ denotes the corresponding perturbation parameter. The common exponent set $(\alpha,\beta,z)=(0,1/2,-2)$ originates from the same coarse-grained mechanism: normal diffusion within a finite accessible domain with a diffusion coefficient that vanishes quadratically at the transition. The agreement between stochastic transport and deterministic chaotic scattering provides strong evidence for a common class of discontinuous dynamical transitions and extends the statistical-mechanics description of phase transitions to integrability-breaking dynamics.

Can the transition from integrability to chaos be discontinuous? We show that it can, and that the resulting first-order dynamical transition coexists with critical slowing down. Using an analytically tractable confined random walk and a deterministic stadium-like billiard, we find a finite jump of the stationary diffusive observable at the transition while the relaxation time diverges. Both systems display normal diffusion and the same exponent set $(\alpha,\beta,z)=(0,1/2,-2)$. We trace this agreement to a common coarse-grained mechanism: diffusion in a finite accessible domain with a diffusion coefficient that vanishes quadratically with the perturbation. The results identify a discontinuous route from integrability to chaos and provide evidence for a broader universality class of first-order dynamical transitions.

We describe how strong vorticity fluctuations appear in three-dimensional incompressible turbulence driven by a large-scale random force. By combining analytical and numerical methods, we construct a vorticity instanton: the most probable force history driving the flow to a prescribed large vorticity at a given point and time. We show how the smooth, large-scale force produces a long vortex filament with a thin, viscous-scale core. Our main finding is that the standard mechanism of amplifying a vortex tube through radial compression and axial stretching is only effective up to a threshold value. Producing vorticity above the threshold requires an additional ingredient: vorticity waves propagating along the tube. The final vorticity is the result of two colliding waves. This mechanism has striking similarities to the one recently suggested by OpenAI for creating infinite vorticity in finite-time Navier–Stokes blow-up.

The deployment of a geoengineering program to cool Earth's climate may be imminent. It is crucial that tools be developed to ensure that such a program would achieve its objectives while minimizing disruption. Here we exploit recently developed differentiable atmospheric models to demonstrate a novel geoengineering control strategy. In the differentiable primitive-equation atmospheric model JAX-GCM we impose a uniform $+4$\,K ocean warming and ask what pattern of sea-surface temperature cooling – in five ocean-masked zonal bands of prescribed SST forcings whose amplitudes are free – returns land near-surface air temperature closest to the model's own unwarmed climatology. This idealized set-up represents a cooling pattern that could be delivered physically either by marine cloud brightening or stratospheric aerosol injection. Gradients through chaotic dynamics decorrelate from the true sensitivity beyond the Lyapunov horizon, so we optimize greedily over segments of 8 to 14 days, following receding-horizon control. The learned strategy removes $92.3 \pm 0.4\%$ of the realized land warming across a ten-member ensemble of two-year rollouts, and a three-year run sustains it. If we use the spatial pattern of land temperature as the optimization objective, the distributions of precipitation, evaporation, and specific humidity over land are restored as well, even though they are not included in the objective function. The learned strategy from JAX-GCM replayed in the AI emulators LUCIE and NeuralGCM without re-optimization is successful, suggesting robustness. These promising results demonstrate a strategy for designing optimal climate interventions that can be applied broadly for geoengineering scenarios under consideration.

A resonator with one localized cubic nonlinearity can acquire additional cusp pairs when it is coupled to an otherwise linear network. At first-harmonic order, the cusp locus is a phase contour of the driving-point receptance $G(\Omega)$ at the nonlinear coordinate. We resolve that contour into the contribution of an active pole and its background. For a lightly damped, isolated modal pole on a slowly varying, nearly real background, cusp-pair birth occurs once the participation of that pole at the nonlinear coordinate exceeds a closed-form threshold set by its damping half-width and that background. On the weak-mixing hardening branch generated by a pole above the primary resonance, the birth coupling then grows as the square root of the damping at fixed mode separation, and of the mode separation at fixed damping. The curvature of the full cusp contour determines whether the local fold set opens a gap (beaks type) or creates an isolated loop (lips type). The isolated-pole approximation reduces this classifier to a background-receptance rule whose range of validity is quantified on 100 random networks. The same ensemble tests the pole ladder numerically, while multi-harmonic continuation follows the whole reconnection sequence—connected band, gap, isolated lens, none—in the original equations at the fold of the birth locus, where the classifier is least protected. The construction also extends to coupled-mode Kerr networks, where the driving-point response sets the intrinsic fold geometry and the drive-port transfer response sets its accessibility.

Fast-flowing glaciers, known as ice streams, control the majority of ice sheet mass loss due to thermomechanical coupling at their base, which can induce basal lubrication and highly variable flow. Even quite simple models can show `binge-purge' oscillations. Recent work has highlighted that coupling between divergent ice streams can also give rise to chaotic variability. In this paper, we return to a model of ice streams with a divergent coupling to identify and correct some physical inconsistencies, most notably a global volume conservation error. We propose and validate a revised Divergent Coupling (DC) model that ensures rigorous mass balance. Bifurcation analysis and Poincaré sections reveal previously unidentified complex frequency-locking (Arnold tongues), a period-adding cascade and chaos associated with torus breakdown. We also consider an equivalent model with a Convergent Coupling (CC), where we find no evidence of chaotic variability. These findings suggest the importance of specific topologies on ice stream variability.

Oscillators defined on curved spaces provide a natural framework for exploring geometry-induced nonlinear dynamics. The two-dimensional Mathews-Lakshmanan oscillator and Higgs oscillator are prominent examples of geometrically induced nonlinear oscillators. The two-dimensional Higgs oscillator in flat space arises from a conformal (stereographic) projection of the harmonic oscillator on the sphere S2 onto the two-dimensional plane. Its one-dimensional counterpart, defined by a non-Euclidean geometry with curvature parameter, exhibits chaotic dynamics and extreme events (EEs) when subjected to damping and external driving forces [1]. In this work, we introduce coupling through mass interaction among multiple one-dimensional damped and forced Higgs oscillators and investigate the resulting collective nonlinear dynamics within a many-particle framework. More importantly, we establish the connection between the conformal mass m(x) and the extrema of velocity through a power-law relation, demonstrating how the conformal mass regulates the turning points leading to the emergence of extreme events (EEs). Furthermore, to provide additional insight into the underlying mechanism, we quantify the energy synchronization error among the oscillators, thereby revealing how energy synchronization evolves during the onset of EEs.

Nonlinearities in multimode mechanical systems can give rise to rich dynamical phenomena with great potential for sensing applications and for future quantum experiments. We demonstrate that the coupled center-of-mass and rotational motion of a Meissner-levitated micromagnet offer a promising platform for nonlinear dynamics, combining low dissipation, magnetic tunability, strong intrinsic Duffing nonlinearities, and nonlinear intermodal couplings. We use this tunability to demonstrate the generation of a mechanical frequency comb in the micromagnet dynamics, realized by parametric excitation of two translational modes followed by cascaded nonlinear frequency mixing, which produces a phononic comb with tunable spacing. At large amplitudes, nonlinear coupling to a low-frequency librational mode of the magnet leads to parametric excitation and phase locking of that mode, generating a dense spectral fine structure at subharmonics of the drive. These results establish levitated micromagnets as a platform for nonlinear multimode mechanics, with potential applications in precision sensing and quantum-limited metrology.

Linear response is cheap to measure; the bistability boundaries it organizes are not. For a passive network with one localized cubic, the driving-point receptance $G$ fixes the period-one cusp set at fundamental-harmonic order: cusps lie on a fixed phase contour of $G$, a tangency of that contour under parameter variation creates a pair, and its curvature separates a gap opening from an isolated loop. For a two-mode absorber the linear prediction locates a benchmark birth coupling to $0.3\%$, and to $0.03\%$ once a third-harmonic correction of scale $|G(3\Omega)/G(\Omega)|$ is included.

In this work, we study finite-time effects in ultracold atomic systems by considering time-dependent modulations with variable waveforms and durations. These two characteristics can be controlled by adjusting only a single parameter. For arbitrarily short pulses, our model recovers the paradigmatic kicked rotor while maintaining the impulse transmitted per period and unit amplitude constant. Furthermore, we demonstrate that finite-time effects have a profound impact on dynamical localization, a result that cannot be captured by the {\delta}-kicked-rotor model. Through a detailed analysis of the effects of different modulation amplitudes, periods, and waveforms, we identify the conditions for which dynamical localization is significantly enhanced. We show that the strength of dynamical localization increases sharply as the system approaches the {\delta}-kicked-rotor limiting case. Moreover, we establish the existence of an optimal value of the period that maximizes dynamical localization for given values of the amplitude and shape parameter.

We develop a framework for offline full-window branch discovery, optionally followed by online continuation with an ensemble Kalman filter (EnKF). Three mechanisms drive the branch search: adjoint path-kernel (APK) differentiation balances kernel differentiation and correction-stabilized path perturbation, shifting the optimization from exploration to exploitation; an optimized Gaussian initial law broadens the search over initial-state basins; and loss-weighted mixing across independent runs recombines successful path components. We may then select an interior state using a local loss and continue online with an EnKF. In 40-dimensional Lorenz-96 experiments, the mean offline path RMSE of APK is 4.3 times smaller than that of population weak-$\mathrm{4D\text{-}Var}_x$. The resulting APK-EnKF method has a mean online RMSE 64 times smaller than that of ordinary EnKF.

The potential of crossing climate tipping points (TP) has reached the attention of many researchers and the general public. On the one hand, the basis for this concern is strengthening, with simulations showing that abrupt transitions might occur even for moderate emission scenarios. On the other hand, our understanding of what constitutes such transitions mathematically is becoming more nuanced. This leads to challenges for the fidelity of early-warning signals (EWS), which accompany bifurcations in systems that closely track a slowly changing steady state. Different kinds of out-of-equilibrium dynamics in a rapidly changing climate, as well as chaos, blur the critical tipping threshold and imply limits in predictability. Using EWS to detect loss of local stability from data requires careful evaluation in the high-dimensional system, and may have limited predictive power in a highly multistable climate, where there is more than a single, well-known alternative state. This calls for new methods using non-equilibrium statistical mechanics and dynamical systems to efficiently probe the global stability properties of climate models and observations.

Data-driven machine learning has established a robust foundation for reconstructing nonlinear dynamical systems from observations, primarily for the purposes of forecasting and control. However, most existing efforts focus on recovering specific observed dynamics rather than the generative synthesis of new ones. Inspired by image fusion and style transfer, we introduce a neural network framework termed Neural Dynamical Transfer Learning (NDTL) to create new systems with prescribed dynamics from pairs of parent nonlinear dynamical systems. By computing fundamental dynamical signatures, including the intrinsic dimension, the Kaplan-Yorke dimension, the invariant measure statistics, and the Lyapunov spectrum, we demonstrate that NDTL preserves key features inherited from the parent models while simultaneously generating novel dynamics. Beyond these validation examples, NDTL induces a criterion for dynamics classification, creates stable oscillatory coexistence in the Hastings-Powell food chain model, produces interpretable epidemiological models, and provides a chaotic source for image encryption.

Symbols can provide compact representations of complex sensory information. Mathematically, chaotic attractors can provide a basis for such representations, with object identity encoded in the topological organization of the attractor. In three-dimensional phase space, this organization can be characterized by the linking matrix of unstable periodic orbits, which serves as a topological fingerprint. Because the linking matrix has no closed-form dependence on the learnable parameters, we learn the chaotic series generating the target attractors rather than optimizing the invariant directly; the reconstructed attractor then determines its unstable periodic orbits and linking matrix through its topology. In this paper, we propose a third-order tensor that maps inputs to connection matrices that generate the corresponding chaotic series. Exploiting the duality between a vector and its sequential representation as a series, we lift inputs into a higher-dimensional space and reorder their components, allowing the input to be presented sequentially rather than simultaneously. We first demonstrate that this representation can discriminate between classes in the \href{https://archive.ics.uci.edu/dataset/151/connectionist+bench+sonar+mines+vs+rocks}{Sonar} dataset. We then examine perceptual constancy using the \href{https://www.csc.kth.se/cvap/databases/kth-tips/index.html}{KTH-TIPS2-a} dataset, where variations in color and texture within a class are mapped to distinct chaotic series associated with the same topological attractor class. The results demonstrate that the proposed tensor-based mapping can preserve attractor class while accommodating variations in the sensory realization of an object.

Quantum reservoir computing (QRC) is a machine learning approach which employs the internal dynamics of a physical system (the reservoir) to encode and process information. In this work, we explore the use of dual-unitary circuits in a brickwork architecture as a platform for QRC, well suited to current noisy intermediate-scale quantum devices. Dual-unitary circuits present both practical and conceptual advantages. Our results indicate that, under appropriate conditions, dual-unitarity can lead to an enhanced regime of operation: we numerically verify that it improves memory effects and nonlinear processing, and shields against finite-shot noise, mitigating exponential concentration. Moreover, dual unitarity offers an intuitive picture of how operator dynamics gives rise to memory and nonlinear processing in circuit-based reservoirs.

We present an analytical, statistical solution to the binary-binary (2+2) outcome of the chaotic non-hierarchical four-body problem. The solution is based on the density-of-states formulation pioneered by J. J. Monaghan. The method skips the computationally expensive integration of the equations of motion, and instead samples the outcome from the chaotic phase-space, subject to conservation of energy, momentum, and angular momentum. From the joint distribution, we extract marginal distributions of several key parameters using Monte-Carlo integration, and numerically verify them by comparing to an identical ensemble of scattering experiments produced by the FEWBODY code. From the comparison, we identify a regime not represented by the density-of-states formulation: the hard binary regime, where one binary is much harder then the four-body energy scale, and the system acts as an effective three-body system. We hypothesize that at this regime the system's probability distribution spreads over an effective reduced three-body chaotic phase space. The process of transfer from four-body to three-body phase-space is still not understood, and represents the next natural extension of density-of-states methods.