Chaos最新文献

Resilience broadly describes the ability to withstand perturbations. Measures of system resilience have gathered increasing attention across applied disciplines; yet, existing metrics often lack computational accessibility and generalizability. In this work, we review the literature on resilience measures through the lens of dynamical systems theory and numerical methods. In this context, we reformulate pertinent measures into a general form and introduce a resource-efficient algorithm designed for their parallel numerical estimation. By coupling these measures with a global continuation of attractors, we enable their consistent evaluation along system parameter changes. The resulting framework is modular and easily extendable, allowing for the incorporation of new resilience measures as they arise. We demonstrate the framework on a range of illustrative dynamical systems, revealing key differences in how resilience changes across systems. This approach offers a more global and comprehensive perspective compared to traditional linear stability metrics used in local bifurcation analysis, which can overlook inconspicuous but significant shifts in system resilience. This work opens the door to genuinely novel lines of inquiry, such as the development of new early warning signals for critical transitions or the discovery of universal scaling behaviors. The presented exemplary analyses can serve as blueprints for further system-specific investigations and comparative studies on different measures of resilience. All code and computational tools are provided as an open-source contribution to the DynamicalSystems.jl software library.

We present spatially localized states in graphene nanoribbons using the simple tight-binding model with nearest neighbor interactions, which correctly describes the electronic properties of graphene close to the Fermi level. By monitoring the time evolution of initially localized wave packets, we identify different final states depending on edge geometry and initial condition. For armchair nanoribbons, we find, both numerically and analytically, flat band states that remain strictly localized across the nanoribbon width instead of spreading in the infinite periodic direction. For zigzag nanoribbons, we find partially flat band states at the Fermi level, which are localized both in the transverse and longitudinal directions, different from the well-known localized edge states that extend along the whole length of the zigzag edge and decay to zero in the transverse direction. The effects of nonlinearity induced by interactions on these states and on wave packet spreading in general are examined within the discrete nonlinear Schrödinger equation model in and out of the self-trapping regime. We also examine the effects of disorder by introducing random on-site energies to find that all wave packets evolve to exponentially localized states, as expected. These localization phenomena with different origin, from edge geometry to nonlinearity and disorder, should affect wave propagation and transport in atomically thin two-dimensional nanostructures and should be observed in honeycomb lattice systems in photonics, cold atoms, and other physical contexts, opening new directions toward the targeted transfer of relevant excitations.

Chaotic systems are crucial for security and signal tasks, but many prior systems need higher dimensions or complex nonlinearities, and most give limited validation of fractional-order numerics and security design. This manuscript investigates a new 3D chaotic system containing an absolute-value nonlinearity. The proposed model exhibits no real equilibria and illustrates interesting robust chaotic behaviors affirmed by bifurcation diagrams, Lyapunov exponent, and sensitivity analysis. We generalize the considered new model to the fractional-order system with aid of the Caputo fractional operator. The Haar wavelet method is utilized to derive the numerical results rigorously for the fractional-order system. We portray its dynamical behavior for different fractional orders to show hidden chaotic dynamics. Phase-space portraits affirm the existence of dissipative chaos even at fractional orders ρ<1. A physics-informed symbolic regression framework is implemented to reformulate governing equations from simulated data, attaining high prediction fidelity. On the top of that, the fractional-order system is utilized to gray scale and red-blue-green image encryption. Extensive cryptographic metrics, such as entropy, number of pixels change rate, unified average changing intensity, and correlation coefficients, verify the strength of the algorithm in resisting statistical and differential attacks. The high dimensionality, structural sensitivity, and parameter-tunable complexity of the model make it a powerful tool for uses in secure communication and nonlinear signal processing.

This study explores the use of fractional-order epidemic models to capture the memory-dependent and nonlinear behaviors inherent in cholera transmission. We present a fractional-order susceptible-infected-recovered-individuals adopting preventive measures-bacteria model that integrates preventive behavior and environmental feedback. By applying the Caputo derivative, we demonstrate the existence, uniqueness, and boundedness of the model's solutions and derive an analytical expression for the basic reproduction number R0. Our stability and bifurcation analyses show how memory influences the system's transition from a disease-free to an endemic state via a forward bifurcation. We also design a fractional optimal control strategy that synthesizes health education, protection, and sanitation measures. Numerical simulations indicate that the fractional dynamics help suppress infection peaks by extending transient memory effects, which enhances the system's resilience to epidemics and lowers environmental contamination. These results underscore the profound impact of fractional-order memory and nonlinear coupling on both epidemic thresholds and the effectiveness of control measures, providing new perspectives on the dynamics of waterborne diseases.

Inferring stochastic dynamics from data is central; yet, in many applications, only unordered, non-sequential measurements are available-often restricted to limited regions of state space-so standard time-series methods fail. We introduce DyNoSeD (Identifying Dynamics from Non-Sequential Data), a first-principles framework that identifies unknown dynamical parameters from such non-sequential data by minimizing Fokker-Planck residuals. We develop two complementary routes: a local route that handles region-restricted data via local score estimation, and a global route that fits dynamics from globally sampled data using a kernel Stein discrepancy without density- or score estimation. When the dynamics are affine-in-the-unknown-parameters (while remaining nonlinear-in-the-state), we prove necessary-and-sufficient conditions for the existence and uniqueness of the inferred parameter vector and derive a sensitivity analysis that identifies which parameters are tightly constrained by the data and which remain effectively free under over-parameterization. For general non-affine parameterizations, both routes define differentiable losses amenable to gradient-based optimization. As demonstrations, we recover (i) the three parameters of a stochastic Lorenz system from non-sequential observations (region-restricted data for the local route and full steady-state data for the global route) and (ii) a 3×7 interaction matrix of a nonlinear gene-regulatory network derived from a published B-cell differentiation model, using only unordered steady-state samples and applying the global route. Overall, DyNoSeD provides two first-principles routes for system identification from non-sequential data, grounded in the Fokker-Planck equation, that link data, density, and stochastic dynamics.

Reservoir computing (RC) is a machine learning framework that uses recurrent neural networks and is characterized by directly capitalizing on intrinsic dynamics instead of adjusting internal parameters. In particular, in the form of physical reservoir computing (PRC), recent studies have advanced by treating various physical systems as reservoirs and applying them to time-series data processing and quantifying information-processing properties. In this way, RC and PRC potentially have interdisciplinary impact, and as more researchers from diverse academic disciplines learn and utilize RC and PRC, there is potential for more creative research to emerge. In this paper, we introduce a Jupyter Notebook-based educational material called RC bootcamp for learning RC, being made publicly available under an open-source license (https://rc-bootcamp.github.io/). The RC bootcamp was originally developed and continuously updated within our research group to efficiently train our collaborators and new students, ultimately enabling them to conduct experiments by themselves. Considering the diverse backgrounds of learners, it starts with the basics of computer science and numerical computation using Python/NumPy, as well as fundamental implementations in RC, such as echo state networks and linear regression. Furthermore, it covers important analytical indicators based on dynamical systems theory, such as Lyapunov exponents, echo state property index, and information-processing capacity, as well as cutting-edge approaches utilizing chaos, including first-order, reduced and controlled error (FORCE) learning and innate training, and attractor design via bifurcation embedding. We expect that the RC bootcamp will become a useful educational material for learning RC and PRC and further invigorate research activities in the RC and PRC fields.

This study explores the complexities of Bazikin's type predator-prey model, incorporating the influence of the mate-finding Allee effect on prey population, the impact of cooperative hunting strategies among predators, and the effect of diffusion. It provides a detailed analysis of how these factors influence ecological interactions and affect species dynamics. A detailed theoretical study is carried out to investigate the possible equilibrium states of the temporal model system. This is followed by an analysis of their stability and instability, along with an in-depth analysis of all possible bifurcation scenarios related to various equilibrium points. This model demonstrates saddle-node, Hopf, and Bogdanov-Takens bifurcations about some model parameters. On the other hand, the positivity and boundedness of solutions of the diffusive model are studied. The dynamics of the diffusive model have been investigated, considering linear as well as non-linear analysis. A qualitative analysis using numerical simulations is performed to validate all analytical findings. Numerical simulations demonstrate the development of diffusion-driven patterns, highlighting the emergence of target patterns, chaotic patterns, spots, stripes, and intricate combinations that merge stripes with spots. The simulation outcomes of the diffusive model indicate that multiple factors, including the predator's attack rate, the Allee effect, cooperative hunting behaviors, and diffusion coefficients, shape spatial distributions. The results of the analysis will help us to explore the relevance of various ecological effects and their impact within biology.

The independent value of the reward and the structural advantages of dynamic networks have been well-established in their respective fields. Yet, research on their integration remains at an exploratory stage. Thus, we effectively incorporated the reward into the dynamic network model to promote cooperation. Numerical simulation results show a clear pattern: When the temptation to defect is low, the cooperation rate stays high regardless of incentive payoff values. However, when the temptation to defect is high, the cooperation rate only remains high if incentive payoff values exceed a certain threshold. These findings demonstrate that the reward can significantly boost cooperation on dynamic networks. The main contributions of this study are threefold: First, to ensure effective integration of the reward into the model, we carefully designed the payoff calculation rule. Second, using the adjusted payoff values, we ingeniously formulated the network-structure evolution rule. Third, through a detailed analysis of the numerical simulation results, we revealed the underlying mechanism behind the improved cooperation levels.

We introduce efficient Network Reciprocity Control (NRC) algorithms for steering the degree of asymmetry and reciprocity in binary and weighted networks while preserving fundamental network properties. Our methods maintain edge density in binary networks and cumulative edge weight in weighted graphs. We test these algorithms on synthetic benchmark networks, including random, small-world, and modular structures, as well as brain connectivity maps (connectomes) from various species. We demonstrate how adjusting the asymmetry-reciprocity balance under edge density and total-weight constraints influences key network features, including spectral properties, degree distributions, community structure, clustering, and path lengths. Additionally, we present a case study on the computational implications of graded reciprocity by solving a memory task within the reservoir computing framework. Furthermore, we establish the scalability of the NRC algorithms by applying them to networks of increasing size. These approaches enable a systematic investigation of the relationship between directional asymmetry and network topology, with potential applications in computational and network sciences, social network analysis, and other fields studying complex network systems where the directionality of connections is essential.

Dengue fever, a major mosquito-borne viral disease, poses a significant public health threat, particularly in high-incidence countries like Brazil, where rising cases strain limited medical resources. We analyze the impact of constrained medical resources, specifically hospital bed capacity, on dengue transmission dynamics. A novel compartmental model is developed where the availability of hospital beds (B) is a key parameter governing treatment access. Employing classical linearization theory, we conduct a comprehensive stability analysis of the system equilibrium points. Systematic bifurcation analysis, utilizing center manifold theory and normal form theory, reveals complex dynamical behaviors: backward bifurcation (indicating disease persistence for basic reproduction numbers R0<1), Saddle-node bifurcation, Hopf bifurcation (giving rise to periodic solutions), and a codimension-2 Bogdanov-Takens bifurcation. Model validation is performed using incidence data from the São Paulo, Brazil dengue outbreak, enabling parameter estimation and calculation of R0. Sensitivity analysis identifies key parameters for disease control. Crucially, hospital bed capacity B exhibits a threshold regulatory effect: below a critical value, backward bifurcation occurs, sustaining endemicity even when R0<1; above the critical value, increasing beds initially reduces infection prevalence, but can subsequently induce periodic oscillations via Hopf bifurcation before further reducing disease burden. This demonstrates that medical resource constraints fundamentally alter epidemic outcomes through nonlinear dynamical mechanisms.