Stationary Probability and First-Passage Time of Biased Random Walk

In this paper, we consider the stationary probability and first-passage time of biased random walk on 1D chain, where at each step the walker moves to the left and right with probabilities p and q respectively(0 p, q 1,p + q = 1). We derive exact analytical results for the stationary probability and first-passage time as a function of p and q for the first time. Our results suggest that the first-passage time shows a double power-law F ^(N-1)~γ, where the exponent γ = 2 for N < |p-q|^(-1) and γ = 1 for N > |p-q|^(-1). Our study sheds useful insights into the biased random-walk process.

Probability Response of Shape Memory Alloy Beam Subjected to Noise Excitation

Shape Memory Alloy(SMA)is a typical material with memory effect,and it is widely used in many engineering fields.Based on the elastic theory and Galerkin method,a vibration system of SMA beam with rigid constraints is proposed.The non⁃smooth transformation was employed to deal with the discontinuous position,and the original system was turned into an approximate equivalent system associated with the Dirac function.Then,using the stochastic averaging method,the drift and diffusion coefficients of the corresponding Fokker Planck Kolmogorov equation were described.Lastly,the approximate probability response of the system was formulated analytically.Meanwhile,numerical simulation was carried out to verify the effectiveness of analytical results.Furthermore,stochastic bifurcation was discussed.Results show that the stationary probability response of the system was affected by the increase of noise amplitude and restitution force,and a certain restitution value and damping could induce P⁃bifurcation.

Stationary response of colored noise excited vibro-impact system

The generalized cell mapping(GCM) method is used to obtain the stationary response of a single-degree-of-freedom.Vibro-impact system under a colored noise excitation. In order to show the advantage of the GCM method, the stochastic averaging method is also presented. Both of the two methods are tested through concrete examples and verified by the direct numerical simulation. It is shown that the GCM method can well predict the stationary response of this noise-perturbed system no matter whether the noise is wide-band or narrow-band, while the stochastic averaging method is valid only for the wide-band noise.

Steady-state analysis subject to a coloured and a white additive of a bistable system multiplicative noise noise with coloured cross-correlated noises

The steady-state properties of a bistable system are investigated when both the multiplicative noise and the coupling between additive and multiplicative noises are coloured with different values of noise correlation times T1 and T2. After introducing a dimensionless parameter R（R = α/D, D is the intensity of the multiplicative noise and a is the intensity of the additive noise）, and performing the numerical computations, we find the following points： （1） For the case of R 〉 1, A （the intensity of correlation between additive and multiplicative noises）, T1 and T2 can induce the stationary probability distribution （SPD） transition from bimodal to unimodal in structure, but for the cases of R _〈 1, the bimodal structure is preserved; （2） a can also induce the SPD transition from bimodal to unimodal in structure; （3） the bimodal structure of the SPD exhibits a symmetrical structure as D increases.

Dynamics and response reshaping of nonlinear predator-prey system undergoing random abrupt disturbances

An actual ecological predator-prey system often undergoes random environmental mutations owing to the impact of natural disasters and man-made destruction, which may destroy the balance between the species. In this paper,the stochastic dynamics of the nonlinear predator-prey system considering random environmental mutations is investigated, and a feedback control strategy is proposed to reshape the response of the predator-prey system against random abrupt environmental mutations. A delayed Markov jump system(MJS) is established to model such a predator-prey system. A novel first integral is constructed which leads to better approximation solutions of the ecosystem. Then, by applying the stochastic averaging method based on this novel first integral, the stochastic response of the predator-prey system is investigated, and an analytical feedback control is designed to reshape the response of the ecosystem from the disturbed state back to the undisturbed one.Numerical simulations finally illustrate the accuracy and effectiveness of the proposed procedure.

Time-Delayed Feedback Control in a Single-Mode Laser System

The effects of time-delayed feedback control in a single-mode laser system is investigated.Using the smalltime delay approximation,the analytic expression of the stationary probability distribution function of the laser field isobtained.The mean,normalized variance and skewness of the steady-state laser intensity are calculated.It is found thatthe time-delayed feedback control can suppress the intensity fluctuation of the laser system.The numerical simulationsare in good agreement with the approximate analytic results.

FEEDBACK CONTROL OPTIMIZATION FOR SEISMICALLY EXCITED BUILDINGS

A feedback control optimization method of partially observable linear structures via stationary response is proposed and analyzed with linear building structures equipped with control devices and sensors. First, the partially observable control problem of the structure under horizontal ground acceleration excitation is converted into a completely observable control problem. Then the It6 stochastic differential equations of the system are derived based on the stochastic averaging method for quasi-integrable Hamiltonian systems and the stationary solution to the Fokker-Plank-Kolmogorov （FPK） equation associated with the It6 equations is obtained. The performance index in terms of the mean system energy and mean square control force is established and the optimal control force is obtained by minimizing the performance index. Finally, the numerical results for a three-story building structure model under E1 Centro, Hachinohe, Northridge and Kobe earthquake excitations are given to illustrate the application and the effectiveness of the proposed method.

Enhancement of density divergence in an insect outbreak model driven by colored noise

The steady states and the transient properties of an insect outbreak model driven by Gaussian colored noise are studied in this paper. According to the Fokker-Planck equation in the unified colored-noise approximation, we analyse the stationary probability distribution and the mean first-passage time of this model. By numerical analysis, the effects of the self-correlation time of insect birth rate and predation rate respectively reveal a manifest population divergence on the insect density. The decrease of the mean first-passage time indicates an enhancement dynamic on the density divergency with colored noise of a large self-correlation time based on the insect outbreak model.

Transitions in a genotype selection model driven by coloured noises

This paper investigates a genotype selection model subjected to both a multiplicative coloured noise and an additive coloured noise with different correlation time τ1 and τ2 by means of the numerical technique. By directly simulating the Langevin Equation, the following results are obtained. （1） The multiplicative coloured noise dominates, however, the effect of the additive coloured noise is not neglected in the practical gene selection process. The selection rate μ decides that the selection is propitious to gene A haploid or gene B haploid. （2） The additive coloured noise intensity and the correlation time τ2 play opposite roles. It is noted that α and τ2 can not separate the single peak, while can make the peak disappear and ^-2 can make the peak be sharp. （3） The multiplicative coloured noise intensity D and the correlation time τ1 can induce phase transition, at the same time they play opposite roles and the reentrance phenomenon appears. In this case, it is easy to select one type haploid from the group with increasing D and decreasing τ1.

Some New Results on Transition Probability

In this paper, we study the basic properties of stationary transition probability of Markov processes on a general measurable space （E, δ）, such as the continuity, maximum probability, zero point, positive probability set,standardization, and obtain a series of important results such as Continuity Theorem, Representation Theorem, Levy Theorem and so on. These results are very useful for us to study stationary tri-point transition probability on a general measurable space （E, δ）. Our main tools such as Egoroff＇s Theorem, Vitali-Hahn-Saks＇s Theorem and the theory of atomic set and well- posedness of measure are also very interesting and fashionable.

On Multivariate Markov Chains for Common and Non-Common Objects in Multiple Networks

Node importance or centrality evaluation is an important methodology for network analysis.In this paper,we are interested in the study of objects appearing in several networks.Such common objects are important in network-network interactions via object-object interactions.The main contribution of this paper is to model multiple networks where there are some common objects in a multivariate Markov chain framework,and to develop a method for solving common and non-common objects’stationary probability distributions in the networks.The stationary probability distributions can be used to evaluate the importance of common and non-common objects via network-network interactions.Our experimental results based on examples of co-authorship of researchers in different conferences and paper citations in different categories have shown that the proposed model can provide useful information for researcher-researcher interactions in networks of different conferences and for paperpaper interactions in networks of different categories.

Tail Asymptotics of Two Parallel Queues with Transfers of Customers

This paper studies a system consisting of two parallel queues with transfers of customers.In the system,one queue is called main queue and the other one is called auxiliary queue.The main queue is monitored at exponential time instances.At a monitoring instant,if the number of customers in main queue reaches L(>K),a batch of L−K customers is transferred from the main queue to the auxiliary queue,and if the number of customers in main queue is less than or equal to K,the transfers will not happen.For this system,by using a Foster-Lyapunov type condition,we establish a sufficient stability condition.Then,we provide a sufficient condition under which,for any fixed number of customers in the auxiliary queue,the stationary probability of the number of customers in the main queue has an exact geometric tail asymptotic as the number of customers in main queue increases to infinity.Finally,we give some numerical results to illustrate the impact of some critical model parameters on the decay rate.