Chaos最新文献

A reduced-order modeling framework is developed to address the high-dimensional challenges of parameterized partial differential equations by integrating tensor-train decomposition (TTD), Gaussian process regression (GPR), and Gaussian process dynamical models (GPDMs). TTD furnishes a low-rank approximation of the solution snapshots, while GPR learns the nonlinear mapping from the input parameter space to the tensor-train format. GPDM then models the temporal dynamics, enabling accurate time evolution prediction and uncertainty quantification. The method is validated on several benchmark problems, including Burgers' equations and the incompressible Navier-Stokes equations. Comparative experiments against traditional methods such as proper orthogonal decomposition-Gaussian process regression and dynamic mode decomposition based on tensor-train decomposition-Gaussian process regression demonstrate that the proposed approach achieves superior accuracy in modeling nonlinear temporal dynamics, conducting time-domain interpolation, and quantifying prediction uncertainty.

Biological neurons are emerging as attractive candidates for artificial intelligence and machine learning applications given their natural energy efficiency and self-repair capacity. However, they differ from their idealized artificial counterparts. Biological neurons have highly variable and noisy dynamics and display intrinsic spontaneous activity instead of purely input-driven dynamics. Moreover, biological neuronal networks have physically constrained and highly plastic connections, leading to a complex and ever evolving connectivity structure. Here, we investigate (numerically and with preliminary experimental data) the stability of the input responses of neuronal cultures using a reservoir computing framework. Utilizing a numerical model for the growth and activity of neuronal cultures, previously used to model experimental data, we investigate the effect of large-scale network topology, specifically homogeneous vs modular architectures, on fading memory, reservoir performance under increasingly noisy dynamics, and robustness to network rewiring. We find that modular networks exhibit longer fading memory time, sustain higher performance under noisy conditions, and are more robust to connectivity rewiring than homogeneous networks. Finally, we observe no relationship between some characteristics of the network adjacency matrix (specifically its spectral properties) and reservoir computing performance.

The Amazon rainforest and the Atlantic Meridional Overturning Circulation (AMOC) are considered to be tipping elements: they are important components of the Earth system, but may collapse under climate change. Moreover, an AMOC collapse may favor the transition of the rainforest to a degraded forest by influencing the precipitation patterns over the Amazon. This phenomenon is known as tipping cascade and better understanding it is key to anticipating the impact of tipping events. Here, we investigate in a coupled conceptual AMOC-Amazon model the probability that an AMOC weakening affects tree cover loss in two regions of the rainforest. To get more insight into the mechanisms behind the tipping cascade, we also analyze the dynamics of both systems and their evolution during the Amazon transition. Namely, we track the transition probability and the transition time of the Amazon and reconstruct the distribution of AMOC strength at every stage of this transition. These tasks require a large ensemble simulation, containing, in particular, a large number of transitions. Since such events may be too rare to be sampled by direct numerical simulation, the collapse of both systems is studied using TAMS (Time Adaptive Multilevel Splitting), a "rare-event" algorithm designed to efficiently sample rare transitions. We find that, in the northwest of Brazil, a transition of the Amazon rainforest to a degraded forest within 200 years is very unlikely. However, in this region, such transition can only occur after an AMOC collapse, which would have a large drying effect that favors the development of extreme wildfires.

We investigate energy propagation in a one-dimensional stub lattice in the presence of both disorder and nonlinearity. In the periodic case, the stub lattice hosts two dispersive bands separated by a flatband; however, we show that sufficiently strong disorder fills all intermediate bandgaps. By mapping the two-dimensional parameter space of disorder and nonlinearity, we identify three distinct dynamical regimes (weak chaos, strong chaos, and self-trapping) through numerical simulations of initially localized wave packets. When disorder is strong enough to close the frequency gaps, the results closely resemble those obtained in the one-dimensional disordered discrete nonlinear Schrödinger equation and Klein-Gordon lattice model. In particular, subdiffusive spreading is observed in both the weak and strong chaos regimes, with the second moment m2 of the norm distribution scaling as m2∝t0.33 and m2∝t0.5, respectively. The system's chaotic behavior follows a similar trend, with the finite-time maximum Lyapunov exponent Λ decaying as Λ∝t-0.25 and Λ∝t-0.3. For moderate disorder strengths, i.e., near the point of gap closing, we find that the presence of small frequency gaps does not exert any noticeable influence on the spreading behavior. Our findings extend the characterization of nonlinear disordered lattices in both weak and strong chaos regimes to other network geometries, such as the stub lattice, which serves as a representative flatband system.

We derive stationary distributions of the so-called mispricing and the trend signal in various regimes of the extended Chiarella model of financial markets. This model is a stochastic nonlinear dynamical system that encompasses dynamical competition between a (saturating) trending and a mean-reverting component. We find the so-called mispricing distribution and the trend distribution to be unimodal Gaussians in the small noise, small feedback limit. Slow trends yield Gaussian-cosh mispricing distributions that allow for a P-bifurcation: unimodality occurs when mean-reversion is fast, bimodality when it is slow. The critical point of this bifurcation is established and refutes previous ad hoc reports and differs from the bifurcation condition of the dynamical system itself. For fast, weakly coupled trends, deploying the Furutsu-Novikov theorem reveals that the result is again unimodal Gaussian. For the same case with higher coupling, we disprove another claim from the literature: bimodal trend distributions do not generally imply bimodal mispricing distributions. The latter becomes bimodal only for stronger trend feedback. The exact solution in this last regime remains unfortunately beyond our proficiency.

This study systematically investigates the synchronization ability of two-layer networks with identical chain structures, with particular focus on the influence of inter-layer edges. For undirected chain networks, we identify the optimal placement of two directed inter-layer edges (co-directional or reverse-directional) that maximize synchronization ability and analyze how synchronization ability varies with inter-layer coupling strength for directed and undirected edges at these optimal positions. Extending to two-layer directed chain networks, we analyze how the number of inter-layer edges (whether directed or undirected) impacts synchronization ability, contrast the coupling-strength-dependent synchronization ability between networks configured with two undirected vs two directed inter-layer edges, and examine the effect of edge positional changes with a fixed number of inter-layer edges. The findings offer theoretical guidance for optimizing synchronization through strategic structural design in multi-layer chain networks.

The synchronization of complex networks, governed by the generalized Fiedler value (γ) of the Laplacian matrix, is critical for functional stability and energy efficiency. However, this property also renders networks vulnerable to targeted disruptions. Traditional percolation-based attack strategies, which focus on structural integrity, often fail to effectively suppress synchronization. This study introduces a Laplacian spectral perturbation approach to systematically identify and remove edges critical to synchronization. By deriving the sensitivity of γ to topological changes and leveraging the gradient of the Fiedler vector, we quantify each edge's contribution to synchronization, revealing its connection to community structure. We propose the Fiedler Gradient Iterative Attack (FGIA) algorithm for static networks, which constructs locally optimal edge-removal sequences to maximize γ degradation while preserving global connectivity. FGIA achieves computational efficiency, outperforming brute-force methods and conventional centrality-based attacks. Extensive simulations on synthetic and real-world networks demonstrate FGIA's superior performance in synchronization suppression, offering practical applications in neuroscience and critical infrastructure protection.

We propose and investigate a generalized Schrödinger equation by introducing a fractional Riesz derivative to account for anomalous transport and a memory kernel to describe temporal nonlocal effects. Additionally, we include a long-range interactions term, modeled by an integral operator, which captures spatially extended interactions. Using the Green function approach, we derive analytical solutions and explore their implications in the time-space domain. Our findings reveal novel quantum phenomena arising from the interplay of fractional dynamics, nonlocal potentials, and memory effects, including the emergence of new local maxima in the evolution of Green's functions and distinct localization behaviors.

Nature's fractal patterns should, in theory, exhibit some common characteristics as revealed by a number of space missions carried out in the neighborhood of Earth's space environment. Here, we show that the overall shape of the multifractal spectrum of galaxies resembles that of NASA's Voyager mission observed at the heliospheric boundaries. We have, therefore, employed the same method-grounded on well-known results from up to one million galaxies in the updated redshift database-to identify a reliable multifractal spectrum of the distribution of galaxies on cosmological scales. We show that the observed spectrum fits the weighted Cantor set, which serves as a template for the turbulence observed in the heliosphere. In the universe, this would be indicative of the galaxy distribution's nonlinear multifractal scaling. For galaxies receding from the Sun at different distances, the degree of multifractality somewhat varies but is smaller than that inside the heliosphere. This could be connected to the presence of voids in the large-scale distribution of matter. Some variations from the Hubble law for the ideal uniform expansion might explain a possible asymmetry of the spectrum. We anticipate that finding nonlinear fractal scaling laws of galaxies will be a major step toward the ultimate explanation of the matter distribution in the Universe, especially fitting on the hundredth anniversary of discovery of the first galaxy beyond the Milky Way.

Walking droplets-millimetric oil droplets that self-propel across the surface of a vibrating fluid bath-exhibit striking emergent statistics that remain only partially understood. In particular, in a variety of experiments, a robust correspondence has been observed between the droplet's statistical distribution and the time-average of the wave field that guides it. Durey et al. [Chaos 28, 096108 (2018)] rigorously established such a correspondence for single-droplet systems with a single, instantaneous droplet-bath impact during each vibration period, but numerical and experimental evidence suggests that the correspondence should hold far more broadly. Laboratory droplet systems, for instance, often exhibit complex bouncing modes that do not adhere to these hypotheses. We attempt to complete this program in the present work, rigorously extending this statistical correspondence to account for arbitrary droplet-bath impact models, multi-droplet interactions, and non-resonant bouncing. We investigate this correspondence numerically in systems of one and two droplets in 1D geometries, and we highlight how the time-averaged wave field can distinguish between correlated and uncorrelated pairs of droplets.