Chaos最新文献

A nonsmooth fold occurs when an equilibrium or limit cycle of a nonsmooth dynamical system hits a switching manifold and collides and annihilates with another solution of the same type. We show that beyond the bifurcation, the leading-order truncation to the system, in general, has no bounded invariant set. This is proved for boundary equilibrium bifurcations of Filippov systems, hybrid systems, and continuous piecewise-smooth ordinary differential equations, and grazing-type events for which the truncated form is a continuous piecewise-linear map. The omitted higher-order terms are expected to be incapable of altering the local dynamics qualitatively, implying the system has no local invariant set on one side of a nonsmooth fold, and we demonstrate this with an example. Thus, if the equilibrium or limit cycle is attracting, the bifurcation causes the local attractor of the system to tip to a new state. The results also help explain global aspects of bifurcation structures of the truncated systems.

Due to the very importance of fractional calculus in studying anomalous stochastic processes, we systematically investigate the existing formulation of fractional calculus and generalize it to broader applied contexts. Specifically, based on the improved Riemann-Liouville fractional calculus operators and the modified Maruyama's notation for fractional Brownian motion, we develop the fractional Ito^'s calculus and derive a generalized Fokker-Planck equation corresponding to the Maruyama's process, along with which, the stochastic realizations of trajectories, both underdamped and overdamped, have been studied in terms of the stochastic dynamics equations newly formulated. This paves a way to study the path integrals and the stochastic thermodynamics of anomalous stochastic processes. We also explicitly derive several fundamental results in fractional calculus, including the relation between fractional and normal differentiation, the Laplace transform for fractional derivatives, the analytic solution of one type of generalized diffusion equations, and the fractional integration formulas. Our results advance the existing fractional calculus and provide practical references for studying anomalous diffusion, mechanics of memory materials in engineering, and stochastic analysis in fractional orders.

In addition to a common synchronization and/or localization behavior, a system of linearly coupled identical bistable van der Pol (BVdP) oscillators can exhibit a "non-conventional" or "modal" synchronization. In a two degree-of-freedom case, one can observe stable beatings attractor with synchronized amplitudes of the symmetric and antisymmetric modes. The current study demonstrates that this unusual behavior is generic and quite ubiquitous, if the system is explored in an appropriate parametric regime. In the absence of the self- excitation, the system of coupled linear oscillators possesses a complete degenerate set of non-interacting eigenmodes. If the system is symmetric and the coupling is weak enough, appropriate initial conditions will result in a continuous multi-parametric family of stationary beatings or beating waves. If the self-excitation terms are small enough, i.e., even weaker than the weak coupling, one can expect that the degeneracy will be removed and stable attractors very close to some special stationary or wave beating responses will appear. This phenomenon is demonstrated in chains of ring-coupled BVdP oscillators. In particular, we observe simple two-wave synchronization for N=2,3,5,6,7 coupled oscillators; for N=2, it corresponds to the modal synchronization observed previously. The case N=4 is special: due to internal resonances in the slow flow, the two-wave synchronization turns unstable, and more complicated patterns of the multi-wave synchronization are revealed. Analytic models manage to capture the shapes of the observed beat waves. All results are verified by direct numeric simulations.

We examine two stochastic processes with random parameters, which in their basic versions (i.e., when the parameters are fixed) are Gaussian and display long-range dependence and anomalous diffusion behavior, characterized by the Hurst exponent. Our motivation comes from biological experiments, which show that the basic models are inadequate for accurate description of the data, leading to modifications of these models in the literature through introduction of the random parameters. The first process, fractional Brownian motion with random Hurst exponent (referred to as FBMRE below) has been recently studied, while the second one, Riemann-Liouville fractional Brownian motion with random exponent (RL FBMRE) has not been explored. To advance the theory of such doubly stochastic anomalous diffusion models, we investigate the probabilistic properties of RL FBMRE and compare them to those of FBMRE. Our main focus is on the autocovariance function and the time-averaged mean squared displacement of the processes. Furthermore, we analyze the second moment of the increment processes for both models, as well as their ergodicity properties. As a specific case, we consider the mixture of two-point distributions of the Hurst exponent, emphasizing key differences in the characteristics of RL FBMRE and FBMRE, particularly in their asymptotic behavior. The theoretical findings presented here lay the groundwork for developing new methods to distinguish these processes and estimate their parameters from experimental data.

Various phenomenological generalizations of the foundational equation in quantum physics have been proposed in prior studies. This paper presents a rigorous analytical derivation, grounded in first principles, that elucidates the impact of quantum fluctuations on the evolution of quantum systems. Furthermore, it demonstrates how this essential generalization can be achieved through statistical methods. The paper reveals that standard linear equations of quantum mechanics are recovered under specific limits of the parameter that governs nonlinear behavior. It establishes a direct correlation between the decay of quantum waves and the magnitude of these fluctuations. This connection provides critical insights into the dynamic properties of quantum systems and their susceptibility to underlying stochastic influences. Moreover, this work successfully formulates a comprehensive approach to a complete family of nonlinear quantum evolution equations. This framework expands our theoretical arsenal and enhances our ability to model and predict the behavior of complex quantum systems under various conditions. This research represents a significant advancement in our understanding of quantum mechanics, offering a more nuanced view of how quantum systems evolve under the influence of intrinsic fluctuations. It paves the way for future explorations into the stability, coherence, and dynamical evolution of quantum states, potentially impacting quantum computing, information processing, and other applications in quantum technology.

From the analytical perspective, we investigate the diffusion processes that arise from a system composed of a surface with a backbone structure coupled to the bulk via the boundary conditions. The problem is formulated in terms of diffusion equations with nonlocal terms, which can be used to model different processes, such as sorption-desorption and reactions on the surface. For the backbone structure, we consider the comb model, which imposes constraints on the diffusion processes in different directions on the surface. The results reveal a broad class of behaviors that can be connected to anomalous diffusion.

Human behavioral awareness is usually socially relevant, decisions are made based on the behavior of individuals, and this dynamic process of human awareness with herd effects is called awareness cascade. Based on the complexity of modern information dissemination, information is not monolithic. Individuals choose the type of epidemic-related information to accept, i.e., whether it is positive or negative information, according to the awareness cascade, and then take the corresponding measures to cope with the epidemic. In this paper, we use the microscopic Markov chain approach to model an information-virus dual network, where the information layer has a threshold model with awareness cascade of positive and negative information, and on the virus layer is a susceptible-infected-recovery model with epidemic infection time delay and recovery time delay. The time delay is also a non-negligible modeling factor as the complete infection of an individual and the complete recovery of an individual require sufficient time. An explicit formula for the critical threshold of epidemic spread for this model is derived. We find that positive and negative information and time delay have a significant effect on the critical threshold, and the recovery time delay is the time delay that mainly affects the epidemic size. Experiments show that the local acceptance rate of positive information has a threshold point for the spread of epidemics under awareness cascade, and that this point is significantly affected by the mass media. The local acceptance rate of negative information also divides the spread of epidemics into two stages.

We consider a generalized SEIS (susceptible, exposed, infectious, and susceptible) model where individuals are divided into three compartments: S (healthy and susceptible), E (infected but not just infectious, or exposed), and I (infectious). Finite waiting times in the compartments yield a system of delay-differential or memory equations and may exhibit oscillatory (Hopf) instabilities of the otherwise stationary endemic state, leading normally to regular oscillations in the form of an attractive limit cycle in the phase space spanned by the compartment rates. In the present paper, our aim is to demonstrate that in the dynamics of delayed SEIS models, persistent chaotic attractors can bifurcate from these limit cycles and become accessible if the nonlinear interaction terms fulfill certain basic requirements, which to our knowledge were not addressed in the literature so far. Computing the largest Lyapunov exponent, we show that chaotic behavior exists in a wide parameter range. Finally, we discuss a more general system and show that a sudden falloff of the infection rate with respect to increasing infection number may be responsible for the emergence of chaotic time evolution. Such a falloff can describe mitigation measures, such as wearing masques, individual isolation, or vaccination. The model may have a wide range of interdisciplinary applications beyond epidemic spreading, for instance, in the kinetics of certain chemical reactions.

In this study, given the inherent nature of dissipation in realistic dynamical systems, we explore the effects of dissipation within the context of fractional dynamics. Specifically, we consider the dissipative versions of two well known fractional maps: the Riemann-Liouville (RL) and the Caputo (C) fractional standard maps (fSMs). Both fSMs are two-dimensional nonlinear maps with memory given in action-angle variables (In,θn), with n being the discrete iteration time of the maps. In the dissipative versions, these fSMs are parameterized by the strength of nonlinearity K, the fractional order of the derivative α∈(1,2], and the dissipation strength γ∈(0,1]. In this work, we focus on the average action ⟨In⟩ and the average squared action ⟨In2⟩ when K≫1, i.e., along strongly chaotic orbits. We first demonstrate, for |I0|>K, that dissipation produces the exponential decay of the average action ⟨In⟩≈I0exp⁡(-γn) in both dissipative fSMs. Then, we show that while ⟨In2⟩RL-fSM barely depends on α (effects are visible only when α→1), any α<2 strongly influences the behavior of ⟨In2⟩C-fSM. We also derive an analytical expression able to describe ⟨In2⟩RL-fSM(K,α,γ).

The post-acute sequelae of COVID-19 (PASC) poses a significant health challenge in the post-pandemic world. However, the underlying biological mechanisms of PASC remain intricate and elusive. Network-based methods can leverage electronic health record data and biological knowledge to investigate the impact of COVID-19 on PASC and uncover the underlying biological mechanisms. This study analyzed territory-wide longitudinal electronic health records (from January 1, 2020 to August 31, 2022) of 50 296 COVID-19 patients and a healthy non-exposed group of 100 592 individuals to determine the impact of COVID-19 on disease progression, provide molecular insights, and identify associated biomarkers. We constructed a comorbidity network and performed disease-protein mapping and protein-protein interaction network analysis to reveal the impact of COVID-19 on disease trajectories. Results showed disparities in prevalent disease comorbidity patterns, with certain patterns exhibiting a more pronounced influence by COVID-19. Overlapping proteins elucidate the biological mechanisms of COVID-19's impact on each comorbidity pattern, and essential proteins can be identified based on their weights. Our findings can help clarify the biological mechanisms of COVID-19, discover intervention methods, and decode the molecular basis of comorbidity associations, while also yielding potential biomarkers and corresponding treatments for specific disease progression patterns.