In this work, we investigate a range of time series, including Gaussian noises (white, pink, and blue), stochastic processes (Ornstein-Uhlenbeck, fractional Brownian motion, and Levy flights), and chaotic systems (the logistic map), using the Visibility Graph (VG) method. We focus on the minimum number of data points required to use VG and on two key descriptors: the degree distribution P(k), which often follows a power law P(k) ~ k^-gamma, and the Hurst exponent H, which identifies persistent and antipersistent time series. While the VG method has attracted growing attention in recent years, its ability to consistently characterize time series from diverse dynamical systems remains unclear. Our analysis shows that the reliable application of the VG method requires a minimum of 1000 data points. Furthermore, we find that for time series with a Hurst exponent H <= 0.5, the corresponding critical exponent satisfies gamma >= 2. These results clarify the sensitivity of the VG method and provide practical guidelines for its application in the analysis of stochastic and chaotic time series.

We consider fluctuating Sabra models of turbulence, which exhibit the phenomenon of spontaneous stochasticity: their solutions converge to a stochastic process in the ideal limit, when both viscosity and small-scale noise vanish. In this paper, we develop a renormalization group (RG) approach to explain this phenomenon. Here, RG is understood as an exact relation between the stochastic properties of systems with different dissipative and noise terms, in contrast to the Kadanoff-Wilson coarse-graining procedure, which involves small-scale integration. We argue that the stochastic process in the ideal limit is represented as a fixed point of the RG operator. The existence of such a fixed point confirms not only the convergence in the ideal limit, but also the universality of the spontaneously stochastic process, i.e. its independence from the type of dissipation and noise. The dominant eigenmode of the linearized RG operator determines the leading correction in the convergence process. The RG eigenvalue $\rho \approx 0.84 \exp(2.28i)$ is universal and it turns out to be complex, which explains the rather slow and oscillatory convergence in the ideal limit. These universality predictions are accurately confirmed by numerical data.

We give the solution to the complete noncommutative Kadomtsev--Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand--Levitan--Marchenko (GLM) equation. This is a linear integral equation in which the scattering data satisfies the linearised KP hierarchy. The solution to the GLM equation is then shown to coincide with the solution to the noncommutative KP hierarchy. We achieve this using two approaches. In the first approach we use the standard Sato-Wilson dressing transformation. In the second approach, which was pioneered by Poppe, we assume the scattering data is semi-additive and by direct substitution, we show that the solution to the GLM equation satisfies the infinite set of field equations representing the noncommutative KP hierarchy. This approach relies on the augmented pre-Poppe algebra. This is a representative algebra that underlies the field equations representing the hierarchy. It is nonassociative and isomorphic to a descent algebra equipped with a grafting product. While we perform computations in the nonassociative descent algebra, the final result which establishes the solution to the complete hierarchy, resides in the natural associative subalgebra. The advantages of this second approach are that it is constructive, explicit, highlights the underlying combinatorial structures within the hierarchy, and reveals the mechanisms underlying the solution procedure.

We propose a general method for constructing a fuzzy cellular automaton from a given cellular automaton. Unlike previous approaches that use fuzzy distinctive normal form, whose update function is restricted to third-order polynomials, our method accommodates a wide range of fuzzification functions, enabling the generation of diverse and complex time-evolution patterns that are unattainable with simpler heuristic models. We demonstrate that phase transitions in pattern formation can be observed by changing the parameters of the fuzzification function or the mixing ratio between two distinct evolution rules of elementary cellular automata. Remarkably, the resulting generalized fuzzy elementary cellular automata exhibit rich dynamical properties, including stable manifolds and chaos, even in minimal systems composed of just three cells.

We define the non-commutative multiple bi-orthogonal polynomial systems, which simultaneously generalize the concepts of multiple orthogonality, matrix orthogonal polynomials and of the bi-orthogonality. We present quasideterminantal expressions for such polynomial systems in terms of formal bi-moments. The normalization functions for such monic polynomials satisfy the non-commutative Hirota equations, while the polynomials provide solution of the corresponding linear system. This shows, in particular, that our polynomial systems form a part of the theory of integrable systems. We study also a specialization of the problem to non-commutative multiple orthogonal polynomials, what results in the corresponding Hankel-type quasideterminantal expressions in terms of the moments. Moreover, such a reduction allows to introduce in a standard way the discrete-time variable and gives rise to an integrable system which is non-commutative version of the multidimensional discrete-time Toda equations.

The question of well-posedness of the generalized Ablowitz-Ladik and Discrete Nonlinear Schrödinger equations with \textit{nonzero} boundary conditions on the infinite lattice is far less understood than in the case where the models are supplemented with vanishing boundary conditions. This question remains largely unexplored even in the standard case of cubic nonlinearities in which, in particular, the Ablowitz-Ladik equation is completely integrable while the Discrete Nonlinear Schrödinger equation is not (in contrast with its continuous counterpart). We establish local well-posedness for both of these generalized nonlinear systems supplemented with a broad class of nonzero boundary conditions and, in addition, derive analytical upper bounds for the minimal guaranteed lifespan of their solutions. These bounds depend explicitly on the norm of the initial data, the background, and the nonlinearity exponents. In particular, they suggest the possibility of finite-time collapse (blow-up) of solutions. Furthermore, by comparing models with different nonlinearity exponents, we prove estimates for the distance between their respective solutions (measured in suitable metrics), valid up to their common minimal guaranteed lifespan. Highly accurate numerical studies illustrate that solutions of the generalized Ablowitz-Ladik equation may collapse in finite time. Importantly, the numerically observed blow-up time is in excellent agreement with the theoretically predicted order of the minimal guaranteed lifespan. Furthermore, in the case of the Discrete Nonlinear Schrödinger equation on a finite lattice we prove global existence of solutions; this is consistent with our numerical observations of the phenomenon of \textit{quasi-collapse}, manifested by narrow oscillatory spikes that nevertheless persist throughout time -- continued in pdf ...

During development, highly ordered structures emerge as cells collectively coordinate with each other. While recent advances have clarified how individual cells process and respond to external signals, understanding collective cellular decision making remains a major challenge. Here, we introduce a minimal, analytically tractable, model of cell patterning via local cell-cell communication. Using this framework, we identify a trade-off between the speed and accuracy of collective pattern formation and, by adapting techniques from stochastic chemical kinetics, quantify how information flows between cells during patterning. Our analysis reveals counterintuitive features of collective patterning: globally optimized solutions do not necessarily maximize intercellular information transfer and individual cells may appear suboptimal in isolation. Moreover, the model predicts that instantaneous information shared between cells can be non-monotonic in time as patterning occurs. An analysis of recent experimental data from lateral inhibition in Drosophila pupal abdomen finds a qualitatively similar effect.

In classical mechanics, driven systems with dissipation often exhibit complex, fractal dynamics known as strange attractors. This paper addresses the fundamental question of how such structures manifest in the quantum realm. We investigate the quantum Duffing oscillator, a paradigmatic chaotic system, using the Caldirola-Kanai (CK) framework, where dissipation is integrated directly into a time-dependent Hamiltonian. By employing the Husimi distribution to represent the quantum state in phase space, we present the first visualization of a quantum strange attractor within this model. Our simulations demonstrate how an initially simple Gaussian wave packet is stretched, folded, and sculpted by the interplay of chaotic dynamics and energy loss, causing it to localize onto a structure that beautifully mirrors the classical attractor. This quantum "photograph" is inherently smoothed, blurring the infinitely fine fractal details of its classical counterpart as a direct consequence of the uncertainty principle. We supplement this analysis by examining the out-of-time-ordered correlator (OTOC), which shows that stronger dissipation clarifies the exponential growth associated with the classical Lyapunov exponent, thereby confirming the model's semiclassical behavior. This work offers a compelling geometric perspective on open chaotic quantum systems and sheds new light on the quantum-classical transition.

Current early warning signs for tipping points often fail to distinguish between catastrophic shifts and less dramatic state changes, such as spatial pattern formation. This paper introduces a novel method that addresses this limitation by providing more information about the type of bifurcation being approached starting from a spatially homogeneous system state. This method relies on estimates of the dispersion relation from noisy spatio-temporal data, which reveals whether the system is approaching a spatially homogeneous (tipping) or spatially heterogeneous (Turing patterning) bifurcation. Using a modified Klausmeier model, we validate this method on synthetic data, exploring its performance under varying conditions including noise properties and distance to bifurcation. We also determine the data requirements for optimal performance. Our results indicate the promise of a new spatial early warning system built on this method to improve predictions of future transitions in many climate subsystems and ecosystems, which is critical for effective conservation and management in a rapidly changing world.

In practical applications, it is crucial that the drive-response systems, although identical in all respects, are synchronized at all times, even if there is noise present. In this work, we test the stability and robustness of three distinct and well-known cryptographic chaotic systems, and compare the results in relation to the desired security.

Understanding and quantifying chaos from data remains challenging. We present a data-driven method for estimating the largest Lyapunov exponent (LLE) from one-dimensional chaotic time series using machine learning. A predictor is trained to produce out-of-sample, multi-horizon forecasts; the LLE is then inferred from the exponential growth of the geometrically averaged forecast error (GMAE) across the horizon, which serves as a proxy for trajectory divergence. We validate the approach on four canonical 1D maps-logistic, sine, cubic, and Chebyshev-achieving R2pos > 0.99 against reference LLE curves with series as short as M = 450. Among baselines, KNN yields the closest fits (KNN-R comparable; RF larger deviations). By design the estimator targets positive exponents: in periodic/stable regimes it returns values indistinguishable from zero. Noise robustness is assessed by adding zero-mean white measurement noise and summarizing performance versus the average SNR over parameter sweeps: accuracy saturates for SNRm > 30 dB and collapses below 27 dB, a conservative sensor-level benchmark. The method is simple, computationally efficient, and model-agnostic, requiring only stationarity and the presence of a dominant positive exponent. It offers a practical route to LLE estimation in experimental settings where only scalar time-series measurements are available, with extensions to higher-dimensional and irregularly sampled data left for future work.

A bootstrap program is presented for algebraically solving the $R$-matrix of a generic integrable quantum spin chain from its Hamiltonian. The Yang-Baxter equation contains an infinite number of seemingly independent constraints on the operator valued coefficients in the expansion of the $R$-matrices with respect to their spectral parameters, with the lowest order one being the Reshetikhin condition. These coefficients can be solved iteratively using Kennedy's inversion formula, which reconstructs the $R$-matrix after an infinite number of steps. For a generic Hamiltonian, the procedure could fail at any step, making the conditions useful as an integrability test. However in all known examples they all turn out to be satisfied whenever the lowest order condition is. It remains to be understood whether they are all implied by the Reshetikhin condition.