Stationary states in bistable system driven by Lévy noise

We study the properties of the probability density function (PDF) of a bistable system driven by heavy tailed white symmetric Lévy noise. The shape of the stationary PDF is found analytically for the particular case of the Lévy index α = 1 (Cauchy noise). For an arbitrary Lévy index we employ numerical methods based on the solution of the stochastic Langevin equation and space fractional kinetic equation. In contrast to the bistable system driven by Gaussian noise, in the Lévy case, the positions of maxima of the stationary PDF do not coincide with the positions of minima of the bistable potential. We provide a detailed study of the distance between the maxima and the minima as a function of the depth of the potential and the Lévy noise parameters.     

Two competing species in super-diffusive dynamical regimes

The dynamics of two competing species within the framework of the generalized Lotka-Volterra equations, in the presence of multiplicative α-stable Lévy noise sources and a random time dependent interaction parameter, is studied. The species dynamics is characterized by two different dynamical regimes, exclusion of one species and coexistence of both, depending on the values of the interaction parameter, which obeys a Langevin equation with a periodically fluctuating bistable potential and an additive α-stable Lévy noise. The stochastic resonance phenomenon is analyzed for noise sources asymmetrically distributed. Finally, the effects of statistical dependence between multiplicative noise and additive noise on the dynamics of the two species are studied.     

非高斯Lévy噪声驱动下的非对称双稳系统的相转移和平均首次穿越时间

研究了非高斯Lévy噪声激励下非对称双稳系统的相转移和首次穿越问题.首先利用Grünwald-Letnikov有限差分方法数值求解系统所对应的分数阶Fokker-Plank方程,得到了系统的稳态概率密度函数.然后分析了系统的非对称参数以及噪声强度和稳定性指标对稳态概率密度函数的影响,发现了非对称参数和稳定性指标的变化都能够诱导系统发生相转移.进一步研究了系统的平均首次穿越时间,得到了非对称参数、噪声强度和稳定性指标影响系统平均首次穿越时间的不同作用机理. 关键词： 非高斯Lévy噪声 非对称双稳系统 相转移 平均首次穿越时间     

Verhulst model with Lévy white noise excitation

The transient dynamics of the Verhulst model perturbed by arbitrary non-Gaussian white noise is investigated. Based on the infinitely divisible distribution of the Lévy process we study the nonlinear relaxation of the population density for three cases of white non-Gaussian noise: (i) shot noise; (ii) noise with a probability density of increments expressed in terms of Gamma function; and (iii) Cauchy stable noise. We obtain exact results for the probability distribution of the population density in all cases, and for Cauchy stable noise the exact expression of the nonlinear relaxation time is derived. Moreover starting from an initial delta function distribution, we find a transition induced by the multiplicative Lévy noise, from a trimodal probability distribution to a bimodal probability distribution in asymptotics. Finally we find a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy stable noise intensity.     

Correlated Lvy flight in external force fields

The correlated Lvy flight is studied analytically in terms of the fractional Fokker-Planck equation and simulated numerically by using the Langevin equation,where the usual white Lvy noise is generalized to an Ornstein-Uhlenbeck Lvy process(OULP)with a correlation timeτc.We analyze firstly the stable behavior of OULP.The probability density function of Lvy flight particle driven by the OULP in a harmonic potential is exactly obtained,which is also a Lvy-type one withτc-dependence width;when the particle is bounded by a quartic potential,its stationary distribution has a bimodality shape and becomes noticeable with the increase of τc.     

Time characteristics of Lévy flights in a steep potential well

Using the method previously developed for ordinary Brownian diffusion, we derive a new formula to calculate the correlation time of stationary Lévy flights in a steep potential well. For the symmetric quartic potential, we obtain the exact expression of the correlation time of steady-state Lévy flights with index α = 1. The correlation time of stationary Lévy flights decreases with an increasing noise intensity and steepness of potential well.     

The analysis of stochastic resonance and bearing fault detection based on linear coupled bistable system under lévy noise

In this paper, the stochastic resonance (SR) phenomenon of the linear coupled bistable system induced by Lévy noise is analyzed. Meanwhile, the characteristics of Lévy noise is also analyzed according to its probability density functions (PDFs) of different stability index α, symmetry parameter β, scale parameter σ and location index μ. The mean of signal-noise ratio increase (MSNRI) is regarded as an index to measure the SR phenomenon. Then, the rules for MSNRI affected by noise intensity D are explored under different charastic indexes of Lévy noise, system parameters a, b, c and coupling coefficient r. The results are beneficial to the numerical simulation of single-frequency and multi-frequency weak signals detection based on single bistable system and linear coupled system respectively. It is found that the performance of the proposed system is better than single bistable system and results of bearing fault detection could also verify the conclusion.     

Metaheuristic optimization-based identification of fractional-order systems under stable distribution noises

This research investigates the identification problem of fractional-order chaotic systems under stable distribution noises. A powerful metaheuristic optimization method called composite differential evolution is used for the identification of the fractional-order Lorenz and Chen systems in the noisy environment, where the structure, parameters, orders and initial values of the systems are all unknown. The identification accuracy is examined when the noise follows the three special cases of stable distributions, i.e., Gaussian, Cauchy and Lévy distributions. In addition, the impact of the four parameters of stable distributions on the identification accuracy is discussed. The experimental results show that the identification error becomes larger when the noise switches from Gaussian to Cauchy and Lévy distributions. The results also turn out that the location of the stable distribution noise plays the most substantial role in the identification accuracy.     

Dynamics and near-optimal control in a stochastic rumor propagation model incorporating media coverage and Lévy noise

The appearance of rumors intensifies people's panic and affects social stability. How to control the spread of rumors has become an important issue which is worth studying. In order to more accurately reflect the actual situation in the real world, a stochastic model incorporating media coverage and Lévy noise is proposed to describe the dynamic process of rumor propagation. By introducing two control strategies of popular science education and media coverage in an emergency event, an near-optimal control problem that minimizes the influence and control cost of rumor propagation is proposed. Sufficient conditions for near-optimal control of the model are established by using a Hamiltonian function. Then the necessary conditions for near-optimal control are obtained by using the Pontryagin maximum principle. Finally, the effect of popular science education, media coverage and Lévy noise on rumor propagation process control is verified by numerical simulation.     

Truncated Lévy flights and generalized Cauchy processes

A continuous Markovian model for truncated Lévy flights is proposed. It generalizes the approach developed previously by Lubashevsky et al. [Phys. Rev. E 79, 011110 (2009); Phys. Rev. E 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010)] and allows for nonlinear friction in wandering particle motion as well as saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and, as shown in the paper, individually give rise to a cutoff in the generated random walks meeting the Lévy type statistics on intermediate scales. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta method. The obtained numerical data were employed to analyze the statistics of the particle displacement during a given time interval, namely, to calculate the geometric mean of this random variable and to construct its distribution function. It is demonstrated that the time dependence of the geometric mean comprises three fragments following one another as the time scale increases that can be categorized as the ballistic regime, the Lévy type regime (superballistic, quasiballistic, or superdiffusive one), and the standard motion of Brownian particles. For the intermediate Lévy type part the distribution of the particle displacement is found to be of the generalized Cauchy form with cutoff. Besides, the properties of the random walks at hand are shown to be determined mainly by a certain ratio of the friction coefficient and the noise intensity rather than their characteristics individually.     

Stochastic resonance in multi-stable system driven by Lévy noise

In this paper, the stochastic resonance (SR) of a multi-stable system driven by Lévy noise is investigated by the mean signal-to-noise ratio gain (SNR-GM). The characteristics for resonant output of multi-stable system, governed by the system parameters (a and c), the noise amplification factor D of Lévy noise are investigated under different values of stability index α and asymmetry parameter β of Lévy noise. The results reveal that the parameter α is closer to 1, the amplitude of SNR-GM versus system parameter a (or c) is larger. The interval of SR presents a trend that the curve of SNR-GM shifts to the right with the increase of α especially when α > 1. In addition, the SNR-GM for different values of system parameter a (or c) exhibits a tendency to move to the left with the increase of system parameter c (or a). Finally, the simulation results prove that the proposed multi-stable model has better advantage than bistable system and monostable system in signal enhancement and SNR-GM performance.     

Quantum Fields Obtained from Convoluted Generalized White Noise Never Have Positive Metric

It is proven that the relativistic quantum fields obtained from analytic continuation of convoluted generalized (Lévy type) noise fields have positive metric, if and only if the noise is Gaussian. This follows as an easy observation from a criterion by Baumann, based on the Dell’Antonio–Robinson–Greenberg theorem, for a relativistic quantum field in positive metric to be a free field.     

Stochastic resonance in underdamped periodic potential systems with alpha stable Lévy noise

In this paper, we investigate the effect of alpha stable Lévy noise with alpha stability index α ($0<\alpha \le 2$) on stochastic resonance (SR) in underdamped periodic potential systems by the non-perturbative expansion moment method and stochastic simulation. Using the spectral amplification factor as a quantifying index, we find that SR can occur in both sinusoidal potentials and ratchet potentials when α is close to 2, while the resonant effect becomes weaker as the stability index decreases. By means of massive numerical statistics, we ascribe this trend to the typical jumps of non-Gaussian Lévy noise ($0<\alpha <2$), which play a destructive role on the periodicity of the long time mean response. We also disclose that the skewness parameter of Lévy noise has a more notable impact on the resonant effect of the asymmetric ratchet potential than that of the symmetric sinusoidal potential because of symmetry breaking.     

On finite truncation of infinite shot noise series representation of tempered stable laws

Tempered stable processes are widely used in various fields of application as alternatives with finite second moment and long-range Gaussian behaviors to stable processes. Infinite shot noise series representation is the only exact simulation method for the tempered stable process and has recently attracted attention for simulation use with ever improved computational speed. In this paper, we derive series representations for the tempered stable laws of increasing practical interest through the thinning, rejection, and inverse Lévy measure methods. We make a rigorous comparison among those representations, including the existing one due to Imai and Kawai [29] and Rosiński (2007) [3], in terms of the tail mass of Lévy measures which can be simulated under a common finite truncation scheme. The tail mass are derived in closed form for some representations thanks to various structural properties of the tempered stable laws. We prove that the representation via the inverse Lévy measure method achieves a much faster convergence in truncation to the infinite sum than all the other representations. Numerical results are presented to support our theoretical analysis.     

Lévy Walk with Multiple Internal States

Lévy walk with multiple internal states can effectively model the motion of particles that don’t immediately move back to the directions or areas which they come from. When the Lévy walk behaves superdiffusion, it is discovered that the non-immediately-repeating property, characterized by the constructed transition matrix, has no influence on the particle’s mean square displacement (MSD) or Pearson coefficient. This is a kind of stable property of Lévy walk. However, if the Lévy walk shows the dynamical behaviors of normal diffusion, then the effect of non-immediately-repeating emerges. For the Lévy walk with some particular transition matrices, it may display nonsymmetric dynamics; in these cases, the behaviors of their variances are detailedly discussed, especially some comparisons with the ones of the continuous time random walks are made (a striking difference is the changes of the exponents of the variances). The first passage time distribution and its average of Lévy walks are simulated, the results of which turn out that the first passage time can distinguish Lévy walks with different transition matrices, while the MSD can not.

     

Lévy flights and Lévy-Schrödinger semigroups

We analyze two different confining mechanisms for Lévy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Lévy-Schrödinger semigroups which induce so-called topological Lévy processes (Lévy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological Lévy process with the same invariant pdf and in reverse.     

Dynamics of a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise

With the development of information technology, rumors propagate faster and more widely than in the past. In this paper, a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise is proposed. The global positivity of the solution process is proved, and further the basic reproductive number R₀ is obtained. When R₀ < 1, the dynamical process of system with Lévy jump tends to the rumor-free equilibrium point of the deterministic system, and the rumor tends to extinction; when R₀ > 1, the rumor will keep spreading and the system will oscillate randomly near the rumor equilibrium point of the deterministic system. The results show that the oscillation amplitude is related to the disturbance of the system. In addition, increasing media coverage can effectively reduce the final spread of rumors. Finally, the above results are verified by numerical simulation.     

Lévy稳定噪声激励下的Duffing-van der Pol振子的随机分岔

研究了Lévy稳定噪声激励下的双稳Duffing-van der Pol振子,利用Monte Carlo方法,得到了振幅的稳态概率密度函数.分析了Lévy稳定噪声的强度和稳定指数对概率密度函数的影响,通过稳态概率密度的性质变化,讨论了噪声振子的随机分岔现象,发现了不仅系统参数和噪声强度可以视为分岔参数,Lévy噪声的稳定指数 α 的改变也能诱导系统出现随机分岔现象. 关键词： Lévy稳定噪声 Duffing-van der Pol振子 稳态概率密度函数 随机分岔     

Schoenberg Correspondence on Dual Groups

As in the classical case of Lévy processes on a group, Lévy processes on a Voiculescu dual group are constructed from conditionally positive functionals. It is essential for this construction that Schoenberg correspondence holds for dual groups: The exponential of a conditionally positive functional is a convolution semigroup of states.     

强束缚势中Lévy飞行的非Gibbs-Boltzmann统计

解析和数值研究了强束缚势中Lévy飞行粒子的稳态分布.结果表明:当势从单稳态变化到双稳态时,粒子的稳态分布呈现单模到双模或双模到三模的转换;特别在势的鞍点处,坐标分布密度函数出现了一个峰,这违背了Gibbs-Boltzmann统计.