加性二值噪声激励下Duffing系统的随机分岔

研究了Duffing系统在加性二值噪声作用下的随机分岔现象.首先,根据二值噪声的统计特性,推导得到二值噪声状态间的跃迁概率,据此对二值噪声进行了数值模拟.其次,利用四阶Runge-Kutta(龙格-库塔)数值算法得到该系统位移和速率的稳态联合概率密度及位移的稳态概率密度.然后,通过对位移稳态概率密度单双峰结构变化的研究,发现加性二值噪声的状态和强度能够诱导系统产生随机分岔现象.最后,观察到随着系统非对称参数的逐渐变化,系统同样产生了随机分岔现象.     

Universality in stochastic enzymatic futile cycle

The enzymatic futile cycle model, which is believe to regulate functional mechanism of bimolecular networks and associated signal processing, is solved analytically within the stochastic framework. The obtained probability distributions of substrate and product at stochastic to deterministic transition exhibit Poisson distribution, and with large ⟨x⟩ limit Normal distribution which are independent of thermodynamic variables indicating universal behavior of molecular distribution. The dynamics of the substrate and product, by simulating the reaction network using stochastic simulation algorithm, exhibit switching mechanism driven by noise in the system. We also observe various distinct noise driven patterns which may correspond to various cellular states. We propose that this noise induce switching mechanism and patterns could be a key element to regulate and control signal processing in molecular networks of cellular system, and may exhibit in the phenotype.     

ON SELECTIVE HARVESTING OF TWO COMPETING FISH SPECIES IN THE PRESENCE OF ENVIRONMENTAL FLUCTUATION

ABSTRACT. The present paper deals with a problem of selective harvesting of two competing fish species in a randomly fluctuating environment. The environmental parameters are assumed to be perturbed by white noise characterized by a Gaussian distribution with mean zero and unit spectral density. The dynamic behavior of the stochastic system is studied and the fluctuations in population are measured both analytically and numerically by computer simulation.     

Optimal Guaranteed Cost Control of Stochastic Discrete-Time Systems with States and Input Dependent Noise Under Markovian Switching

A problem of robust guaranteed cost control of stochastic discrete-time systems with parametric uncertainties under Markovian switching is considered. The control is simultaneously applied to both the random and the deterministic components of the system. The noise (the random) term depends on both the states and the control input. The jump Markovian switching is modeled by a discrete-time Markov chain and the noise or stochastic environmental disturbance is modeled by a sequence of identically independently normally distributed random variables. Using linear matrix inequalities (LMIs) approach, the robust quadratic stochastic stability is obtained. The proposed control law for this quadratic stochastic stabilization result depended on the mode of the system. This control law is developed such that the closed-loop system with a cost function has an upper bound under all admissible parameter uncertainties. The upper bound for the cost function is obtained as a minimization problem. Two numerical examples are given to demonstrate the potential of the proposed techniques and obtained results.     

Dynamically Active Compartments Coupled by a Stochastically Gated Gap Junction

We analyze a one-dimensional PDE-ODE system representing the diffusion of signaling molecules between two cells coupled by a stochastically gated gap junction. We assume that signaling molecules diffuse within the cytoplasm of each cell and then either bind to some active region of the cell’s membrane (treated as a well-mixed compartment) or pass through the gap junction to the interior of the other cell. We treat the gap junction as a randomly fluctuating gate that switches between an open and a closed state according to a two-state Markov process. This means that the resulting PDE-ODE is stochastic due to the presence of a randomly switching boundary in the interior of the domain. It is assumed that each membrane compartment acts as a conditional oscillator, that is, it sits below a supercritical Hopf bifurcation. In the ungated case (gap junction always open), the system supports diffusion-induced oscillations, in which the concentration of signaling molecules within the two compartments is either in-phase or anti-phase. The presence of a reflection symmetry (for identical cells) means that the stochastic gate only affects the existence of anti-phase oscillations. In particular, there exist parameter choices where the gated system supports oscillations, but the ungated system does not, and vice versa. The existence of oscillations is investigated by solving a spectral problem obtained by averaging over realizations of the stochastic gate.     

Electron dynamical response in InP semiconductors driven by fluctuating electric fields

The complexity of electron dynamics in low-doped n-type InP crystals operating under fluctuating electric fields is deeply explored and discussed. In this study, we employ a multi-particle Monte Carlo approach to simulate the non-linear transport of electrons inside the semiconductor bulk. All possible scattering events of hot electrons in the medium, the main details of the band structure, as well as the heating effects, are taken into account. The results presented in this study derive from numerical simulations of the electron dynamical response to the application of a sub-Thz electric field, fluctuating for the superimposition of an external source of Gaussian correlated noise. The electronic noise features are statistically investigated by computing the correlation function of the velocity fluctuations, its spectral density and the variance, i.e. the total noise power, for different values of amplitude and frequency of the driving field. Our results show the presence of a cooperative non-linear behavior of electrons, whose dynamics is strongly affected by the field fluctuations. Moreover, the electrons self-organize among different valleys, giving rise to the reduction of the intrinsic noise. This counterintuitive effect critically depends on the relationship among the characteristic times of the external fluctuations and the temporal scales of complex phenomena involved in the electron dynamical response. In particular, the correlation time of the electric field fluctuations appears to be crucial both for the noise reduction effect and the appearance of an anomalous diffusion effect.     

On noise modeling in a nerve fibre

We present a novel mathematical approach to model noise in dynamical systems. We do so by considering the dynamics of a chain of diffusively coupled Nagumo cells affected by noise. We show that the noise in a variable representing the transmembrane current can be effectively modeled as fluctuations in the model parameters corresponding to electric resistance and capacitance of the membrane. These fluctuations may account for the interactions between the membrane and the surrounding (physiological) solution as well as for the thermal effects. The proposed approach to model noise in a nerve fibre is an alternative to the standard technique based on the consideration of additive stochastic current perturbation (the Langevin type equations) and differs from it in important mathematical aspects, particularly, it points out to the non-Markov dynamics of transmembrane potential. Our scheme relates to a time scale which is shorter than the relaxation times of involved physiological processes.     

Nonequilibrium statistics of a reduced model for energy transfer in waves

We study energy transfer in a “resonant duet”—a resonant quartet where symmetries support a reduced subsystem with only 2 degrees of freedom—where one mode is forced by white noise and the other is damped. We consider a physically motivated family of nonlinear damping forms and investigate their effect on the dynamics of the system. A variety of statistical steady states arise in different parameter regimes, including intermittent bursting phases, states highly constrained by slaving among amplitudes and phases, and Gaussian and non-Gaussian quasi-equilibrium regimes. All of this can be understood analytically using asymptotic techniques for stochastic differential equations. © 2006 Wiley Periodicals, Inc.     

Jump Phenomena and Bifurcations in Stochastic Vehicle-Road Dynamics

When vehicles ride on uneven roads, they are excited to vertical random vibrations whose stationary rms-values (root-mean-square) strongly depend on the velocity of the vehicle. To investigate this vibration behavior, it is appropriate to introduce road models in way domain which are based on the theory of stochastic differential equations and transformed from way to time by means of velocity-dependent way and noise increments. The random base excitations by roads are applied to nonlinear quarter car models. They lead to stationary rms-values of the vertical vehicle vibrations which become resonant for critical velocities and show jump phenomena similar to those of the Duffing oscillator under harmonic excitations. In the stochastic case, jump phenomena are only observable for narrow-banded road excitations. They vanish for increasing car damping and excitation bandwidth. For efficient simulations of the road-vehicle model, the n state equations are utilized to derive n(n + 1)/2 stochastic covariance equations. For small step sizes, their numerical mean square solutions coincide with the nonlinear results of fix-point iterations obtained when the noise terms of the covariance equations are omitted. It can easily be shown, that this deterministic approach leads to the correct stationary covariances in the linear case. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic model of phase transition and metastability

The evolution of a system with phase transition is simulated by a Markov process whose transition probabilities depend on a parameter. The change of the stationary distribution of the Markov process with a change of this parameter is interpreted as a phase transition of the system from one thermodynamic equilibrium state to another. Calculations and computer experiments are performed for condensation of a vapor. The sample paths of the corresponding Markov process have parts where the radius of condensed drops is approximately constant. These parts are interpreted as metastable states. Two metastable states occur, initial (gaseous steam) and intermediate (fog). The probability distributions of the drop radii in the metastable states are estimated. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 94–106, April, 2000.     

A mathematical framework for stochastic climate models

There has been a recent burst of activity in the atmosphere‐ocean sciences community in utilizing stable linear Langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction. Here a systematic mathematical strategy for stochastic climate modeling is developed, and some of the new phenomena in the resulting equations for the climate variables alone are explored. The new phenomena include the emergence of both unstable linear Langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances. All of these phenomena are derived from a systematic self‐consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit. The theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated. The feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography. Explicit concrete examples with the new phenomena are presented for the stochastically forced three‐mode interaction equations. The conjecture of Smith and Waleffe [Phys. Fluids 11 (1999), 1608–1622] for stochastically forced three‐wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach. Examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self‐consistency of the mathematical approach with the predictions of equilibrium statistical mechanics. In particular, for these examples, the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables. © 2001 John Wiley & Sons, Inc.     

Noise-induced Transitions for an SIV Epidemic Model with Medical-resource Constraints

We consider a stochastically forced epidemic model with medical-resource constraints. In the deterministic case, the model can exhibit two type bistability phenomena, i.e., bistability between an endemic equilibrium or an interior limit cycle and the disease-free equilibrium, which means that whether the disease can persist in the population is sensitive to the initial values of the model. In the stochastic case, the phenomena of noise-induced state transitions between two stochastic attractors occur. Namely, under the random disturbances, the stochastic trajectory near the endemic equilibrium or the interior limit cycle will approach to the disease-free equilibrium. Besides, based on the stochastic sensitivity function method, we analyze the dispersion of random states in stochastic attractors and construct the confidence domains (confidence ellipse or confidence band) to estimate the threshold value of the intensity for noise caused transition from the endemic to disease eradication.     

Simulated test of electric activity of neurons by using Josephson junction based on synchronization scheme

The chaotic circuit of resistive–capacitive–inductive-shunted Josephson junction is used to simulate behavior of Hindmarsh–Rose neuronal discharges. Based on tracking control theory, the controller contains two gain coefficients was constructed to control the chaotic system of Josephson junction to synchronize the chaotic Hindmarsh–Rose system, and the single controller was approached analytically. The results confirmed that the controller with appropriate gain coefficients was effective to reach complete synchronization (the amplitudes and rhythms of two systems are identical), phase synchronization (rhythms of two systems are identical) of Josephson junction and Hindmarsh–Rose neurons, respectively. The power consumption is estimated in a feasible way. As a result, the electric activities of Hindmarsh–Rose neurons could be simulated by using Josephson junction model completely.     

The metastable behavior of the three-dimensional stochastic Ising model I

The metastable behavior of the stochastic Ising model in a finite three-dimensional torus is studied in the limit as the temperature goes to zero. All metastable states are characterized and a hierarchic structure is found. For a large class of initial states, the logarithmic asymptotics of the hitting time of the states are studied with all spins +l or − 1. Project supported in part by the State Education Commission of China, the National Natural Science Foundation of China, the Tianyuan Foundation and the National 863 Project.     

How Brownian displacement goes down during actin-myosin interaction in the laser trap

Several laboratories have successfully used laser trap technology to observe and score nanometre-size displacements of an actin filament produced by a single myosin molecule. Given the molecular nature of the measurements, the influence of Brownian noise is significant. We use a Langevin-type stochastic model, together with a two-state Huxley kinetics for myosin attachment to and detachment from actin, to describe the magnitude of displacement (fluctuations) for the attached and detached states. When myosin is attached to the actin filament system, we derive an effective (composite) stiffness, which produces a 26% noise reduction compared to the original Brownian noise. With two model mechanisms discussed, the preferred model mechanism predicts a mean myosin-induced actin filament displacement of 8–9 nm, all of which are in very good agreement with recent laser trap experimental observations.     

Robust Control of Markovian Switching Stochastic Systems with Noise Dependent States and Inputs

A problem of state feedback stabilization of discrete-time stochastic processes under Markovian switching and random diffusion (noise) is considered. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the rate vector and the diffusion term. Sufficient conditions based on linear matrix inequalities (LMI's) for stochastic stability is obtained. The robustness results of such stability concept against all admissible uncertainties are also investigated. An example is given to demonstrate the obtained results.     

Gradual degradation in the supercurrent state of a chaotic superconducting tunnel junction under time-varying perturbation

We report on the degradation of the zero-voltage supercurrent generated in a Josephson tunnel junction residing in an asymmetric potential of the ratchet type, and driven by a quasiperiodic external signal having incommensurable frequencies with irrational ratio ω₂/ω₁ equals to the Golden Mean. In the underdamped regime and via computing the current–voltage (I–V) characteristic curves, we demonstrate that the disappearance of the superconducting state can be correlated to chaotic behaviour, where dynamical phase fluctuations and symmetry breakings associated with the potential and modulating signal are substantially taking place.     

Remarks on stochastic permanence of population models

This paper examines the asymptotic behavior of the stochastic Lotka–Volterra model under Markovian switching. We show that the stochastic Lotka–Volterra model is stochastically permanent. Moreover, we give another type of stochastic permanence, and consider the relationship of two types of stochastic permanence under white noise perturbation.     

Horseshoe-shaped maps in chaotic dynamics of long Josephson junction driven by biharmonic signals

A collective coordinate approach is applied to study chaotic responses induced by an applied biharmonic driven signal on the long Josephson junction influenced by a constant dc-driven field with breather initial conditions. We derive a nonlinear equation for the collective variable of the breather and a new version of the Melnikov method is then used to demonstrate the existence of Smale horseshoe-shaped maps in its dynamics. Additionally, numerical simulations show that the theoretical predictions are well reproduced. The subharmonic Melnikov theory is applied to study the resonant breathers. Results obtained using this approach are in good agreement with numerical simulations of the dynamics of the Poincaré islands.     

Stochastic Modelling of Wind Speed Power Production Correlations

Based on measurements we investigate the velocity-power characteristic of a 2 MW wind turbine. We apply a stochastic analysis where we describe the evolution of the power output with a Langevin equation, with special respect to short-time fluctuations in wind speed. Standard procedures, such as the IEC 61400-12 standard, are limited due to the fact that only mean values over several minutes of wind speed and power output are considered. According to this, short-time dynamics of wind and power fluctuations are usually not taken into account. We introduce an improved method which enables us to extract these dynamics of the power characteristic from the measured data. In particular, we get the response dynamics of the power L (u (t)) via the estimation of Kramers-Moyal coefficients, describing its evolution in time (t) with a Langevin equation where we separate the power output into a relaxation and a noise part. A fixed-point analysis provides the required power characteristic. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)