Papers

Fluctuation–response relations (FRRs) provide a fundamental relation between spontaneous fluctuations and the linear response to external perturbations, yet their validity in non-equilibrium systems remains an open problem. In a general class of noisy, finite-size, mean-field models of coupled phase oscillators, we reveal two types of FRRs in the incoherent phase below the synchronization transition. The first, corresponding to the infinite-size limit (I-FRR), involves the convolution between the external force and the correlation function in the time domain. The second, the finite-size counterpart (F-FRR), is governed by finite-size fluctuations and carries a correction factor: a spectrum function whose roots correspond to eigenvalues, or Landau poles, of the linear operator. The finite-size fluctuations are suppressed in the weak-coupling or strong-noise limits, and F-FRR reduces to I-FRR. Our analytical findings are verified by extensive numerical simulations.

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.

This work introduces a spectral transfer formalism for nonlinear synchronisation dynamics to resolve how internal dynamical activity redistributes across network scales, moving beyond traditional global observables like the Kuramoto order parameter. By projecting Kuramoto phase dynamics onto the eigenbasis of the graph Laplacian, we construct a time-dependent transfer matrix and derive novel network metrics, including the spectral flux, which quantifies directional redistribution between structural dynamical scales rather than physical transport across nodes. Numerical simulations on modular and hierarchical networks reveal highly structured, intermittent spectral transfer episodes characterised by forward and inverse spectral cascades, directional reversals, and complex interaction geometries. Crucially, we demonstrate a clear separation between macroscopic coherence and microscopic spectral interactions, showing that highly organised, fluctuating internal redistribution dynamics persist even beneath nearly stationary and featureless global synchronisation states. This framework extends conventional synchronisation analysis and establishes a multiscale approach for interpreting network dynamics as an evolving process of spectral reorganisation.

Although integrability does not universally require a continuous one-site symmetry, we rigorously prove that every jointly analytic, regular Yang-Baxter deformation of the qutrit exchange interaction necessarily retains a nontrivial, analytically varying one-site charge. Breaking this local symmetry imposes a fundamental physical constraint on approximate conservation, governed by the optimal uniform bound $\delta^3 \le C\varepsilon$ that explicitly relates the minimal one-site symmetry defect $\delta$ to the local current-conservation residual $\varepsilon$. While breaking all one-site charges strictly forbids an exact integrable completion, an optimally compensated nearest-neighbor interaction saturates this cubic limit and anomalously extends the guaranteed infinite-temperature energy-current correlation window to order $|\lambda|^{-3}$ in the perturbation strength $\lambda$. Furthermore, we reveal a fundamental resonance obstruction for intrinsic conversion perturbations that strictly prevents any exact first-order repair of a broken one-site charge on any finite ring. Nevertheless, we demonstrate that the complete eight-dimensional charge memory matrix remains thermodynamically protected and approaches the identity for timescales $t=o(|\lambda|^{-3/2})$, a robust feature of the full infinite-temperature dynamics when the thermodynamic limit is taken before weak coupling.

The danger of collective risk in a public goods game could provide an effective escape route from the tragedy of the commons. Existing studies usually embed risk mitigation in the contribution itself, whereas many real systems rely on a separate costly and non-excludable protective activity that requires a participation quorum. Whether such protection can evolve and persist remains unclear. Here, we formulate an $N$-player public-goods game with ordinary cooperators, defectors, and protective cooperators. Defectors endogenously increase a saturating probability of collective failure, whereas protective cooperators incur an additional cost and reduce this risk for all group members once their number reaches the quorum. Replicator analysis and numerical bifurcation calculations yield three main findings. First, protective cooperation gains an advantage when the focal participant completes the quorum and the resulting expected loss reduction exceeds its additional cost. This advantage varies non-monotonically with group composition. Second, when a single protective cooperator cannot activate protection, protection effectiveness does not enter its first-order invasion fitness at full defection; greater effectiveness cannot overcome the rare-invasion barrier when full defection is locally asymptotically stable. Third, a boundary saddle–node bifurcation creates a saddle and a locally asymptotically stable coexistence equilibrium of defectors and protective cooperators. When full cooperation and full defection are also locally asymptotically stable, three attractors coexist, and the numerically observed evolutionary outcome depends on the initial population composition. These results distinguish the effectiveness of activated protection from its ability to become established from rarity.

General expressions are derived for the amplitude equation valid at a Turing bifurcation of a system of reaction-diffusion equations in one spatial dimension, with an arbitrary number of components. The normal form is computed up to fifth order, which enables the detection and analysis of codimension-two points where the criticality of the bifurcation changes. The expressions are implemented within a Python package, in which the user needs to specify only expressions for the reaction kinetics and the values of diffusion constants. The code is augmented with a Mathematica routine to compute curves of Turing bifurcations in a parameter plane and automatically detect codimension-two points. The software is illustrated with examples that show the versatility of the method including a case with cross-diffusion, a higher-order scalar equation, and a four-component system.

A wave bifurcation is the counterpart to a Turing instability in reaction-diffusion systems, but where the critical wavenumber corresponds to a pure imaginary pair rather than a zero temporal eigenvalue. Such bifurcations require at least three components and give rise to patterns that are periodic in both space and time. Depending on boundary conditions, these patterns can comprise either rotating or standing waves. Restricting to systems in one spatial dimension, complete formulae are derived for the evaluation of the coefficients of the weakly nonlinear normal form of the bifurcation up to order five, including those that determine the criticality of both rotating and standing waves. The formulae apply to arbitrary $n$-component systems ($n\geq 3$) and their evaluation is implemented in software which is made available as supplementary material. The theory is illustrated on two different versions of three-component reaction-diffusion models of excitable media that were previously shown to feature super- and subcritical wave instabilities and on a five-component model of two-layer chemical reaction. In each case, two-parameter bifurcation diagrams are produced to illustrate the connection between complex dispersion relations and different types of Hopf, Turing, and wave bifurcations, including the existence of several codimension-two bifurcations.

Necessary and sufficient conditions are provided for a diffusion-driven instability of a stable equilibrium of a reaction-diffusion system with $n$ components and a diagonal diffusion matrix. These can be either Turing or wave instabilities. Known necessary and sufficient conditions are reproduced for there to exist diffusion rates that cause a Turing bifurcation of a stable homogeneous state in the absence of diffusion. The method of proof here though, which is based on a study of dispersion relations in the contrasting limits in which the wavenumber tends to zero and to $\infty$, gives a constructive method for choosing diffusion constants. The results are illustrated on a model for the dispersion of malaria, a 3-component FitzHugh-Nagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4 components

We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at $q=\varepsilon$, where $\varepsilon$ is a primitive $N$th root of unity with $N$ odd. For the commuting layer transfer matrices $\mathbb{T}(y)$ with mixed boundary conditions, we establish the quantum Frobenius property \[ \mathbb{T}(y)\mathbb{T}(\varepsilon y)\cdots \mathbb{T}(\varepsilon^{N-1}y) = \mathscr{P}_N(y^N)\mathbb{I}, \] where $\mathscr{P}_N$ is a scalar Laurent polynomial obtained directly from the monomial expansion of $\mathbb{T}(y)$. Our proof exploits the underlying three-dimensional integrability, in particular the tetrahedron equations and their consequences. The quantum Frobenius relation essentially determines the joint eigenvalues of the associated Hamiltonians up to multiplicities and shows that they take a free-parafermion form, thereby extending such structures from previously known one-dimensional examples to a broad class of genuinely two-dimensional quantized six-vertex models. The construction also admits independent local gauge choices and recovers the long-range-interacting $\tau_{\; 2}$ model, which includes the original free-parafermion model as a special case. We also apply the same framework to the relativistic quantum Toda chain at odd roots of unity and obtain the free-parafermion spectra for the relevant Q6V transfer matrices with fixed boundary states.

We study the spectral stability of bright and dark solitons in the discrete nonlinear Schrödinger (DNLS) equation using multiprecision arithmetic. The eigenvalues governing stability are exponentially small in the lattice spacing and cannot be resolved with standard double precision. To address this, we develop a computational framework combining multiprecision arithmetic, an exact Jacobian for the stationary problem, and a squared-operator formulation for spectral analysis. This enables accurate resolution of exponentially small eigenvalues and direct comparison with exponential-asymptotic predictions. Our results show that onsite bright solitons are spectrally stable, whereas intersite bright solitons and both onsite and intersite dark solitons are unstable. Bright solitons require only a few eigenvalues and allow efficient large-scale computations, while dark solitons demand higher precision due to their proximity to the continuous spectrum. Simulations up to \(N=65{,}250\) grid points (31.7 GB RAM) highlight the necessity of multiprecision arithmetic for capturing beyond-all-orders spectral effects.

Self-organization of nonlinear waves underpins phenomena ranging from hydrodynamic pattern formation to optical frequency-comb generation. We introduce and numerically demonstrate a shift-register synchronization mechanism for pulse trains in quadratic nonlinear resonators. The synchronized state persists despite substantial group-velocity mismatch through continuous parametric energy exchange, resulting in deterministic discrete pulse shifts after each cavity round trip. Nonlinear parametric interaction modifies the effective walk-off, allowing the pulse trains to satisfy the shift-register synchronization conditions. The proposed mechanism reveals a previously unexplored synchronization regime in quadratic resonators and provides a route toward multicolour frequency combs, optical memories, and ultrafast photonic information processing.

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.

We systematically investigate breather solutions in the modified Korteweg-de Vries (mKdV) equation via numerical simulations. We show that the breather solutions are stable under a variety of perturbations, including amplitude changes, modifications of internal parameters, and perturbations of the nonlinearity of the equation. Our results are not only consistent with the known analytical studies on the stability of breathers in integrable systems, extending them further, but also provide numerical evidence that such breather-type structures can persist and remain stable over the simulated time scales in some non-integrable settings with symmetric potentials.

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.

We study the dispersionless limit of the Manakov system, the integrable two-component generalization of the nonlinear Schrödinger equation. We derive the resulting four-component genus-zero Manakov-Whitham system, characterize its hydrodynamic structure, and show that it passes the Haantjes tensor test for integrability. We show that the branch points of the spectral curve associated with plane wave solutions of the Manakov system are the local Riemann invariants of the dispersionless system. We also use the characteristic speeds to classify the baseband modulational stability/instability of the plane waves, and we study a direct linearization of the Manakov system to characterize their finite-wavenumber stability and verify agreement with the Whitham prediction in the long-wave limit. Finally, we validate the predictions by comparing them with the results of direct numerical simulations.

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.

Adaptive regulation can itself perturb the state it is intended to stabilize. In replicated simulations of an adaptive agent, we compare external disturbance, persistent internally generated disturbance, and control-generated disturbance under regulation-first and disturbance-first ordering. Persistent internal disturbance produces the largest exposure and regulatory burden within the tested parameter grid. When positive controller updates generate an immediate disturbance cost, increasing that cost produces a nonmonotonic response: effective disturbance initially rises, variability across stochastic runs increases over an intermediate range, and corrective activity becomes strongly suppressed at higher costs. The results show how disturbance source and timing shape exposure and controller burden in this model. They motivate testing adaptive agents with distinct disturbance sources and assessing regulatory activity alongside exposure.

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.

The idea that emotions in biological systems are analogous to intrinsic colored noise is advanced and justified. A model describing the dynamics of operation of biological networks under the influence of colored noise is suggested. The agents of a biological network can be represented either by biological species, such as humans and animals, or by neurons of the brain, or by the nodes of a neural network. Operational actions, or decisions, in a biological noisy network are based not only on the evaluation of utility of alternatives, but also on the agents emotions. The model is probabilistic, with the choice of alternatives characterized by the related probabilities. At the initial step, the agents make decisions individually and then start exchanging information with each other and imitating the actions of other agents, thus forming a biological network of interacting agents. Numerical simulations are accomplished for a heterogeneous society consisting of three groups of agents, one group possessing long-range memory, the other group, short-range memory, and the third group of super-rational agents acting strictly on the basis of utility, being deprived of emotions. Dynamics of opinions in different groups can be smooth, oscillatory, or chaotic. Altogether, eight types of operation are found, depending on the dynamics of group decisions. Under strong imitation effect, there appears chaotic motion in the evolution of decision choice. It is shown how the Ellsberg paradox can be resolved and how the exchange of information influences the dynamics of this paradox.

Probably not. For Airy-type initial data, although nearly all lobes accelerate, the principal lobe remains essentially stationary. Moreover, the overall wave packet gradually disperses.

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.

Previous studies of network dynamics have suggested that heterogeneity among nodes inhibits stability, at odds with the ubiquity of inherently heterogeneous natural and engineered systems. Here, we show that this conclusion arises from model reductions introduced for mathematical tractability and breaks down when nodal dynamics are higher-dimensional, yielding non-Hermitian Jacobians. In such systems, including neural, power-grid, and material networks, nodal heterogeneity can instead enhance stability, even when parameters are randomly disordered. Non-Hermiticity also underlies the stabilizing effects of network heterogeneity, which can arise even in one-dimensional nodal dynamics through nonreciprocal interactions, as shown for ecological networks. Our framework reveals disorder not as a liability but as a general resource for stabilizing complex systems.

We introduce causal graph rewriting, a model of computation in which local rules are applied on directed acyclic graphs in an asynchronous manner. The non-determinism arising from asynchrony is disciplined by the oriented edges, which must be understood as both computational dependencies and locality constraints—and are themselves subject to the rewriting. We illustrate the model through two examples: a particle system, and a time-dilation example—reminiscent of general relativity. We study the well-definedness and properties of induced subgraphs and graph composition, which isolate and recombine the region affected by a rewrite. We then study locality with respect to these constructions, showing how a local rewrite preserves positions, borders, and context. Our main result concerns sequential composition: locality extends from single rule applications to arbitrary valid sequences, as any local rule is automatically $*$-local and $*$-extensive. We also formalise and prove the simulation of any one-dimensional cellular automaton.

A composite soliton can lose its binding through a change in the ordering of the vacua that fill its interior and its exterior, while the quantum correction to its mass stays small. We show this in the non-degenerate double sine-Gordon model, normal ordered at the meson mass of the primary vacuum. At one loop the central secondary vacuum becomes the true vacuum at a coupling of order the classical splitting of the vacua, because the quantum tension of a string of secondary vacuum carries the logarithm of the meson-mass ratio. Matrix-product-state calculations for $n=4$ confirm the inversion, extrapolate to the continuum, and are reproduced by the Gaussian effective potential. Finite-chain calculations at two lattice spacings show the string joining the two halves of the $Q=4$ multi-kink lengthening as its tension falls and, past the crossing, filling the box, with the slope of the energy in the box length equal to the vacuum-energy difference and the correction to the mass still under a tenth of the classical value.

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.

We report on frequency bursting oscillations in a system of phase oscillators with adaptive and delayed coupling. Adaptation of the coupling strengths is considered slow and depends on the phase shift between the oscillators. We find due to the combined chain of adaptation, collective dynamics, and time delays, the system robustly achieves a state in which the oscillator's frequencies are nearly synchronized but detuned by an integer number of small adaptation frequencies. We demonstrate that this quantization of the detuning is caused by alternating slow and fast transitions. Moreover, the observed motions take the form of bursts of instantaneous frequency, and the number of spikes in each burst corresponds to the quantization level of the detuning. We provide a fast-slow analysis of this phenomenon and explain the mechanisms behind the emergence of bursts. Our findings indicate that these frequency bursting oscillations are robust and exist stably within finite parameter regions.