Chaos最新文献

Time-evolving graphs arise frequently when modeling complex dynamical systems such as social networks, traffic flow, and biological processes. Developing techniques to identify and analyze communities in these time-varying graph structures is an important challenge. In this work, we generalize existing spectral clustering algorithms from static to dynamic graphs using canonical correlation analysis to capture the temporal evolution of clusters. Based on this extended canonical correlation framework, we define the spatiotemporal graph Laplacian and investigate its spectral properties. We connect these concepts to dynamical systems theory via transfer operators and illustrate the advantages of our method on benchmark graphs by comparison with existing methods. We show that the spatiotemporal graph Laplacian allows for a clear interpretation of cluster structure evolution over time for directed and undirected graphs.

This paper examines the circumstances under which a one-degree-of-freedom approximate system can be employed to predict the dynamics of a cantilever beam comprising an elastic element with a significant mass and a concentrated mass embedded at its end, impacting a moving rigid base. A reference model of the system was constructed using the finite element method, and an approximate lowest-order model was proposed that could be useful in engineering practice for rapidly ascertaining the dynamics of the system, particularly for predicting both periodic and chaotic motions. The number of finite elements in the reference model was determined based on the calculated values of natural frequencies, which were found to correspond to the values of natural frequencies derived from the application of analytical formulas. The precision of the parameter identification and the outcomes yielded by the substitute model were validated through the calculation of the regions of stable periodic solutions using the analytical Peterka method. Subsequently, the qualitative and quantitative limits of the substitute model's applicability were determined. The quantitative limits were delineated through the utilization of Lyapunov exponents and characteristics associated with the energy dissipation due to impacts and the average number of impacts per excitation period. These characteristics provide a foundation for the introduction of global distance measures of the dynamic behavior of diverse systems within a specified range of the control parameter.

We investigate the dynamics of the adaptive Kuramoto model with slow adaptation in the continuum limit, N→∞. This model is distinguished by dense multistability, where multiple states coexist for the same system parameters. The underlying cause of this multistability is that some oscillators can lock at different phases or switch between locking and drifting depending on their initial conditions. We identify new states, such as two-cluster states. To simplify the analysis, we introduce an approximate reduction of the model via row-averaging of the coupling matrix. We derive a self-consistency equation for the reduced model and present a stability diagram illustrating the effects of positive and negative adaptation. Our theoretical findings are validated through numerical simulations of a large finite system. Comparisons of previous work highlight the significant influence of adaptation on synchronization behavior.

Complex network approaches have been emerging as an analysis tool for dynamical systems. Different reconstruction methods from time series have been shown to reveal complicated behaviors that can be quantified from the network's topology. Directed recurrence networks have recently been suggested as one such method, complementing the already successful recurrence networks and expanding the applications of recurrence analysis. We investigate here their performance for the analysis of nonlinear and complex dynamical systems. It is shown that there is a strong parallel with previous Markov chain approximations of the transfer operator, as well as a few differences explained by their structure. Notably, the spectral analysis provides crucial information on the dynamics of the system, such as its complexity or dynamical patterns and their stability. Possible advantages of the directed recurrence network approach include the preserved data resolution and well defined recurrence threshold.

This study explores the impact of stochastic resetting on the random walk dynamics within scale-free (u,v)-flowers. Utilizing the generating function technique, we develop a recursive relationship for the generating function of the first passage time and establish a connection between the mean first passage time with and without resetting. Our investigation spans multiple scenarios, with the random walker starting from various positions and aiming to reach different target nodes, allowing us to identify the optimal resetting probability that minimizes the mean first passage time for each case. We demonstrate that stochastic resetting significantly improves search efficiency, especially in larger networks. These findings underscore the effectiveness of stochastic resetting as a strategy for optimizing search algorithms in complex networks, offering valuable applications in domains such as biological transport, data networks, and search processes where rapid and efficient exploration is vital.

Numerical study of periodic windows for the logistic map is carried out. Accurate rigorous bounds for periodic windows' end points are computed using interval arithmetic based tools. An efficient method to find the periodic window with the smallest period lying between two other periodic windows is proposed. The method is used to find periodic windows extremely close to selected points in the parameter space and to find a set of periodic windows to minimize the maximum gap between them. The maximum gap reached is 4×10-9. The phenomenon of the existence of regions free from low-period windows is explained.

Investment in resources is essential for facilitating information dissemination in real-world contexts, and comprehending the influence of resource allocation on information dissemination is, thus, crucial for the efficacy of collaborative networks. Nonetheless, current studies on information dissemination frequently fail to clarify the complex interplay between information distribution and resources in network contexts. In this work, we establish a resource-based information dissemination model to identify the complex interplay by examining the propagation threshold and equilibriums. We assess the model's efficacy by juxtaposing the mean-field method with Monte Carlo simulations across three author collaboration networks. In addition, we define the function of resources in information dissemination and evaluate the model's applicability using propagating threshold, time evolution, and parametric analyses. Our findings indicate that an increase in available resources accelerates and expands the distribution of information. Notably, we identify abrupt transition phenomena concerning available resources and demonstrate that the information self-learning rate and the information review rate hasten this transition, while information decline and re-diffusion rates decelerate it.

Porous earth materials exhibit large-scale deformation patterns, such as deformation bands, which emerge from complex small-scale interactions. This paper introduces a cross-diffusion framework designed to capture these multiscale, multiphysics phenomena, inspired by the study of multi-species chemical systems. A microphysics-enriched continuum approach is developed to accurately predict the formation and evolution of these patterns. Utilizing a cellular automata algorithm for discretizing the porous network structure, the proposed framework achieves substantial computational efficiency in simulating the pattern formation process. This study focuses particularly on a dynamic regime termed "cross-diffusion wave," an instability in porous media where cross-diffusion plays a significant role-a phenomenon experimentally observed in materials like dry snow. The findings demonstrate that external thermodynamic forces can initiate pattern formation in cross-coupled dynamic systems, with the propagation speed of deformation bands primarily governed by cross-diffusion and a specific cross-reaction coefficient. Owing to the universality of thermodynamic force-flux relationships, this study sheds light on a generic framework for pattern formation in cross-coupled multi-constituent reactive systems.

This study introduces a five-compartment model to account for the impacts of vaccination-induced recovery and nonlinear treatment rates in settings with limited hospital capacity. To reflect real-world scenarios, the model incorporates multiple reinfections in both vaccinated and recovered groups. It reveals a range of dynamics, including a disease-free equilibrium and up to six endemic equilibria. Notably, the model demonstrates that COVID-19 can persist even when the basic reproduction number is less than one, due to backward bifurcation, which conditions the global stability of the disease-free equilibrium. Various bifurcations are analyzed, including saddle-node, Bogdanov-Takens of codimension-2, and Hopf bifurcation of codimension-1. As transmission rates increase, unstable oscillations stabilize, with the Hopf bifurcation becoming supercritical. The model also highlights forward hysteresis, driven by the multistability of endemic equilibria. Key factors influencing the disease's local endemic behavior, such as effective transmission rates and reinfection rates among vaccinated and recovered individuals, are emphasized. Numerical simulations validate the model and underscore its practical relevance.

The dynamics of the convergence for the stationary state considering a Duffing-like equation are investigated. The driven potential for these dynamics is supplied by a damped forced oscillator that has a piecewise linear function. Fixed points and their basins of attraction were identified and measured. We used entropy basin techniques to characterize the basins of attraction, where a changeover in its boundary basin entropy is observed concerning the boundary length. Additionally, we have a set of polar coordinates to describe the asymptotic convergence of the dynamics based on the range of the control parameter and initial conditions. The entire convergence to the stationary state was characterized by scaling laws.