Correlated Gaussian White Noises Research Articles

Dynamic Characteristics of Bistable Vibration Energy Harvester Incorporating Electromagnetic Generator

Although various vibration energy harvesters have been designed over the past few decades, efforts to develop efficient, broadband energy harvesters continue. This work provides a detailed insight into a bistable vibration harvester subjected to correlated Gaussian white noise, with the friction between the rack and pinion described by the slip-stick model. Using the harmonic balance method, the frequency response curve of the amplitude under different mass ratios is discussed. The system response will be enhanced with an increased mass ratio for sinusoidal excitation, but not in the case of random excitation. By employing the stochastic average of the energy envelope, the dynamical governing equation of the harvester is solved, and the probability density functions (PDFs) under different damping coefficients, nonlinear stiffness of the restoring force, and excitation intensities are derived. The results are compared with those from Monte Carlo simulations (MCS) and show good accuracy. The results reveal the presence of P-bifurcations. When the nonlinear stiffness and damping coefficient vary, the number of peaks in the PDFs of system displacement and velocity changes. By adjusting the system parameters, the motion of the system can be significantly enhanced.

Deterministic and Stochastic Research of Cubic Population Model with Harvesting

A detailed examination of the effect of harvesting on a population has been carried out by extending the standard cubic deterministic model by considering a population under Allee effect with a quadratic function representing harvesting. Weak and strong Allee effect transitions, carrying capacity, and Allee threshold change according to harvesting are first discussed in the deterministic model. A Fokker–Planck equation has been obtained starting from a Langevin equation subject to correlated Gaussian white noise with zero mean, and an Approximate Fokker–Planck Equation has been obtained from a Langevin equation subject to correlated Gaussian colored noise with zero mean. This allowed to calculate the stationary probability distributions of populations, and thus to discuss the effects of linear and nonlinear (Holling type-II) harvesting for populations under Allee effect and subject to white and colored noises, respectively.

Probabilistic analysis on parametric random vibration of a marine riser excited by correlated Gaussian white noises

A path integration method is used to discuss the stochastic response of a marine riser excited parametrically and externally by correlated Gaussian white noises. On the basis of the nonlinear vibration equation of a marine riser, the first-order vibration mode is studied and its governing equation is formulated. The response probability density function (PDF) is further developed by a path integration method. During the solution procedure, the Gaussian closure method and the short-time Gaussian approximation method are used to develop the transition probability density function. The results of different time intervals are compared with the results of Monte-Carlo simulation (MCS) to determine the adequate time interval. After that, the path integration method with a Gauss–Legendre scheme is used to obtain the stationary PDF solution of displacement and velocity. The path integration solutions (PIS) are compared with those of the equivalent linearization method (EQL) and MCS. Five demonstrative cases are analyzed to evaluate the effectiveness of the path integration method. The influences of nonlinearity and excitations are considered. It shows that the results of PIS coincide better with the results of MCS than those of EQL. The analysis shows that parametric excitation causes the softening behavior of the PDF distribution compared with those of EQL. The increase of excitation correlation results in the nonsymmetrical PDF distribution of response. The increase of nonlinearity in displacement can lead the PDF of displacement to showing a hardening behavior.

Complex Dynamics of Credit Risk Contagion with Time-Delay and Correlated Noises

The stochastic time-delayed system of credit risk contagion driven by correlated Gaussian white noises is investigated. Novikov’s theorem, the time-delay approximation, the path-integral approach, and first-order perturbation theory are used to derive time-delayed Fokker-Planck model and the stationary probability distribution function of the dynamical system of credit risk contagion in the financial market. Using the method of numerical simulation, the Hopf bifurcation and chaotic behaviors of credit risk contagion are analyzed when time-delay and nonlinear resistance coefficient are varied and the effects of time-delay, nonlinear resistance and the intensity and the correlated degree of correlated Gaussian white noises on the stationary probability distribution of credit risk contagion are investigated. It is found that, as the infectious scale of credit risk and the wavy frequency of credit risk contagion are increased, the stability of the system of credit risk contagion is reduced, the dynamical system of credit risk contagion gives rise to chaotic phenomena, and the chaotic area increases gradually with the increase in time-delay. The nonlinear resistance only influences the infectious scale and range of credit risk, which is reduced when the nonlinear resistance coefficient increases. In addition, the curve of the stationary probability distribution is monotone decreasing with the increase in parameters value of time-delay, nonlinear resistance, and the intensity and the correlated degree of correlated Gaussian white noises.

Response probability density functions of Duffing–Van der Pol vibro-impact system under correlated Gaussian white noise excitations

This paper aims to investigate the stationary probability density functions (PDFs) of a Duffing–Van der Pol vibro-impact system excited by correlated Gaussian white noise. With the help of non-smooth transformation, the stationary PDFs are formulated analytically by the stochastic averaging of energy envelope. The analytical results are verified by numerical simulation results. Stochastic bifurcations for different parameters are considered, and several special PDF forms are observed in this paper. The first form is the shape of the PDF of total energy can be similar to a crater, which has a minimum and a maximum. The second one is the shape of the joint PDF with three peaks, that is to say, the section of joint PDF has three maximum and two minimum. In addition, the influence of the degree of the correlation of Gaussian white noises is explored.

Effect of Correlated Noises in a Genetic Model

The Stratonovich stochastic differential equation is used to analyze genotype selection in the presence of correlated Gaussian white noises. We study the steady state properties of the genotype selection and discuss the effects of the correlated noises. It is found that the degree of correlation of the noises can be used to select one type of genes from another type of mixing genes. The strong selection of genes caused by a large value of multiplicative noise intensity can be weakened by the intensive negative correlation.

Delay induced transitions in an asymmetry bistable system and stochastic resonance

The delay Fokker-Planck equation is given for an asymmetry bistable system with correlated Gaussian white noises. The small delay approximation based on the probability density approach is used and the approximate stationary probability density function is obtained. The phenomenon of delay induced transitions is found. When a weak periodic signal is added, the phenomenon of stochastic resonance is investigated. Expression of the signal-to-noise ratio (SNR) is obtained by using the two-state theory. It is shown that the time delay can suppress or promote the stochastic resonance phenomenon.

Stochastic time-delayed systems driven by correlated noises: Steady-state analysis

In the present paper, a class of stochastic time-delayed systems driven by correlated Gaussian white noises are considered. Novikov’s theorem is used to derive delay Fokker–Planck equations. In the case of small time delays, approximate stationary solutions are obtained. As a special case, the delay Fokker–Planck equation and the approximate stationary probability density function is obtained for a bistable system. Numerical simulations show that the small-delay approximation is a good approximation in the case of small delay. Furthermore, the effects of the correlated noises and the feedback are investigated. The critical curve separating the unimodal and bimodal regions of the stationary probability distribution is shown to be affected by λ (the degree of the correlation of the noises) and ε (the strength of the feedback). Both λ and ε can change the curve of the stationary probability density function from a bimodal to a unimodal structure.

Time-delayed Logistic system driven by correlated Gaussian white noises

Time-delayed Logistic system driven by correlated Gaussian white noises could be a more true reflection of the growth of tumor cell population. In this paper, we study the Logistic system through the small time delay approximation to obtain the probability density of the system in its steady state, and further analyze the influences of the additive and multiplicative noise intensity, the noise-correlated time and the time delay on the steady-state probability density. The non-equilibrium phase transition phenomena induced by the noises is also studied.

Noise in Genotype Selection Model**The project supported by National Natural Science Foundation of China (Grant No. 10275099) and Natural Science Foundation of Guangdong Province of China (Grant Nos. 021707 and 001182)

We study the steady state properties of a genotype selection model in presence of correlated Gaussian white noise. The effect of the noise on the genotype selection model is discussed. It is found that correlated noise can break the balance of gene selection and induce the phase transition which can makes us select one type gene haploid from a gene group.

Correlated noise in a logistic growth model.

The logistic differential equation is used to analyze cancer cell population, in the presence of a correlated Gaussian white noise. We study the steady state properties of tumor cell growth and discuss the effects of the correlated noise. It is found that the degree of correlation of the noise can cause tumor cell extinction.

Effect on the mean first passage time in symmetrical bistable systems by cross-correlation between noises

We present an analytic investigation of the mean first passage time in two opposite directions (from the left well to the right well and from right to left) by studying symmetrical bistable systems driven by correlated Gaussian white noises, and prove that the mean first passage time in two opposite directions is not symmetrical any more when noises are correlated. As examples, the mean first passage time in the quartic bistable model and the sawtooth bistable model are calculated, respectively. From the analytic results of the mean first passage time, we testify further the relation T( from x − to x +,λ)≠T( from x + to x −,λ) in the same area of the parameter plan. Moreover, it is found that the dependences of T + (i.e., T( from x − to x +,λ) ) and T − (i.e., T( from x + to x −,λ) ) upon the multiplicative noise intensity Q and the additive noise intensity D exhibit entirely different properties. For same areas of the parameter plan: in the quartic bistable system, when the T + vs. Q curve exhibits a maximum, while the T − vs. Q curve is monotonous; when the T + vs. D curve is monotonous, while the T − vs. D curve experiences a phase transition from decreasing monotonously to possessing one minimum. Increasing Q, when the T + vs. D curve experiences a phase transition from decreasing monotonously to possessing one maximum, while the T − vs. D curve only increases monotonously. Similar behaviours also exist in the sawtooth bistable model.

Stochastic multiresonance in a bistable sawtooth potential driven by correlated multiplicative and additive noise

We present an analytic investigation of the signal-to-noise ratio (SNR) by studying the bistable sawtooth system driven by correlated Gaussian white noises. The analytic expression of SNR is obtained. Based on it, we detect the phenomenon of stochastic multiresonance, which arises from the dependence of SNR upon the noises correlation coefficient. Furthermore, there exists not only resonance, but also suppression in the SNR∼D (the additive noise intensity) curve and the SNR∼Q (the multiplicative noise intensity) curve.

Random Response to Periodic Excitation with Correlated Disturbances

The paper addresses non-Gaussian stationary response of linear single-degree-of-freedom (SDOF) systems subject to a periodic excitation with correlated random amplitude and phase disturbances that are modeled as correlated Gaussian white noise processes. Correlation between amplitude and phase modulation is specified by the cross-correlation coefficient. Numerical results for the second and fourth moment responses are presented. The probability density function of the response is calculated based on the cumulant-neglect closure method. Non-Gaussian nature of the response is discussed in terms of the excess factor. The results show that the moment responses generally increase with larger random amplitude disturbance and may decrease with larger random phase modulation for a lightly damped system at resonance. The cross correlation between amplitude and phase disturbances plays an important role in the system moment response. Larger system damping results in smaller system moment responses. The moment response may approach a limiting value, depending on the intensity of the amplitude disturbance, as the relative detuning or phase modulation increases. For the case of the phase modulation alone, the response may become Gaussian in the sense of up to the fourth-order moment for sufficiently large relative detuning or random phase disturbances.

Bistable kinetic model driven by correlated noises: Steady-state analysis.

A simple rule to obtain the Fokker-Planck equation for a general one-dimensional system driven by correlated Gaussian white noises is proved by the functional method. The Fokker-Planck equation obtained in this paper is applied to the bistable kinetic model. We find the following for the steady state. (1) In the \ensuremath{\alpha}-D parameter plane (\ensuremath{\alpha} is the strength of the additive noise and D is the multiplicative noise strength), the critical curve separating the unimodal and bimodal regions of the stationary probability distribution (SPD) of the model is shown to be affected by \ensuremath{\lambda}, the degree of correlation of the noises. As \ensuremath{\lambda} is increased, the area of the bimodal region in the \ensuremath{\alpha}-D plane is contracted. (2) When we take a point fixed in the \ensuremath{\alpha}-D plane and increase \ensuremath{\lambda}, the form of the SPD changes from a bimodal to a unimodal structure. (3) The positions of the extreme value of the SPD of the model sensitively depend on the strength of the multiplicative noise, and weakly depend on the additive noise strength. (4) For \ensuremath{\lambda}=1, the case of perfectly correlated noises, when the parameters \ensuremath{\alpha} and D take values in the neighborhood of the line \ensuremath{\alpha}=D in the \ensuremath{\alpha}-D plane, the SPD's corresponding to the points \ensuremath{\alpha}/Dg1 and \ensuremath{\alpha}/D1 exhibit a very different shape of divergence. Therefore, the ratio \ensuremath{\alpha}/D=1 plays the role of a ``critical ratio.''

Environmental stochasticity of gompertz model with time delay

We consider the environmental stochasticity of the Gompertz model with time delay when the parameters are assumed to be described by correlated Gaussian white noise processes. The delay kernel is assumed to be exponentially decaying and the Fokker-Planck equation is obtained in both the Stratonovich and Ito calculi. The exact expressions for the first and second order moments of the logarithm of population size are evaluated and the stability of the system in mean as well as In mean-square is discussed.