The Estimates of the Mean First Exit Time of a Bistable System Excited by Poisson White Noise

Xu, Yong; Li, Hua; Wang, Haiyan; Jia, Wantao; Yue, Xiaole; Kurths, Jürgen

doi:10.1115/1.4037158

Cited by 61 publications

(12 citation statements)

References 33 publications

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“…It has been successfully applied to analyze the stochastic behaviors of the ecosystem under stochastic continuous excitations [18,19,22,37]. In recent years, the stochastic averaging method has been generalized to investigate the nonlinear system with random jump excitations [38][39][40][41]. In the following section, we will employ the stochastic averaging method to study stochastic dynamics of the ecosystem (7).…”

Section: Stochastic Modelmentioning

confidence: 99%

Stochastic Dynamics of a Time-Delayed Ecosystem Driven by Poisson White Noise Excitation

Jia

2018

Entropy

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We investigate the stochastic dynamics of a prey-predator type ecosystem with time delay and the discrete random environmental fluctuations. In this model, the delay effect is represented by a time delay parameter and the effect of the environmental randomness is modeled as Poisson white noise. The stochastic averaging method and the perturbation method are applied to calculate the approximate stationary probability density functions for both predator and prey populations. The influences of system parameters and the Poisson white noises are investigated in detail based on the approximate stationary probability density functions. It is found that, increasing time delay parameter as well as the mean arrival rate and the variance of the amplitude of the Poisson white noise will enhance the fluctuations of the prey and predator population. While the larger value of self-competition parameter will reduce the fluctuation of the system. Furthermore, the results from Monte Carlo simulation are also obtained to show the effectiveness of the results from averaging method.

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Section: Stochastic Modelmentioning

confidence: 99%

Stochastic Dynamics of a Time-Delayed Ecosystem Driven by Poisson White Noise Excitation

Jia

2018

Entropy

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“…Comparing (14) and (15), such transformation function ( 1 , 2 , ) can be written by the following equation:…”

Section: Problem Formulationmentioning

confidence: 99%

“…The eigenvalue problem for a differential operator of three independent variables 1 , 2 , and will be identified from (16), in which Λ( ) is the eigenvalue and ( 1 , 2 , ) is the associated eigenfunction. Meanwhile, the eigenvalue Λ( ) is seen to be the Lyapunov exponent of the th moment of system (1) from (15). Applying the perturbation theory, both the moment Lyapunov exponent Λ( ) and the eigenfunction ( 1 , 2 , ) are expanded in power series of ; that is,…”

Section: Moment Lyapunov Exponentsmentioning

confidence: 99%

“…And then Meng et al [11,12] obtained a sufficient condition of uniform stability for nonlinear integrodifferential system. However, when we consider the dynamical behavior of the system in a real environment, the stochastic excitation cannot be ruled out, such as wind gusts, earthquakes, and ocean waves [13][14][15]. Stochastic stability of the system under the stochastic excitation has attracted more and more attention [16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

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Stochastic Stability Analysis of Coupled Viscoelastic Systems with Nonviscously Damping Driven by White Noise

Liu

Xie

2018

Mathematical Problems in Engineering

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Nonviscously damped structural system has been raised in many engineering fields, in which the damping forces depend on the past time history of velocities via convolution integrals over some kernel functions. This paper investigates stochastic stability of coupled viscoelastic system with nonviscously damping driven by white noise through moment Lyapunov exponents and Lyapunov exponents. Using the coordinate transformation, the coupled Itô stochastic differential equations of the norm of the response and angles process are obtained. Then the problem of the moment Lyapunov exponent is transformed to the eigenvalue problem, and then the second-perturbation method is used to derive the moment Lyapunov exponent of coupled stochastic system. Lyapunov exponent also can be obtained according to the relationship between moment Lyapunov exponent and Lyapunov exponent. Finally, the effects of various physical quantities of stochastic coupled system on the stochastic stability are discussed in detail. These results are validated by the direct Monte Carlo simulation technique.

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“…For the dynamic systems without time-delay, Wang et al investigated the stochastic resonance in a FitzHugh-Nagumo model with an additive Lévy noise numerically; the numerical simu-lation results show the occurrence of the stochastic resonance phenomena in the given FHN system [12]. Xu et al proposed a method to find an approximate theoretical solution to the mean first exit time of a one-dimensional bistable kinetic system subjected to additive Poisson white noise and derived the analytical solution to the mean first exit time by combining perturbation techniques with Laplace integral method [13]. Xu et al discussed the constructive role of combined harmonic and random excitation on stochastic resonance (SR) in a Brusselator model; the simulation results showed that the intensity of the Gaussian colored noise and the amplitude of the periodic force can enhance SR [14].…”

Section: Introductionmentioning

confidence: 99%

Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

Zhang

et al. 2019

Adv Differ Equ

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The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

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The Estimates of the Mean First Exit Time of a Bistable System Excited by Poisson White Noise

Cited by 61 publications

References 33 publications

Stochastic Dynamics of a Time-Delayed Ecosystem Driven by Poisson White Noise Excitation

Stochastic Dynamics of a Time-Delayed Ecosystem Driven by Poisson White Noise Excitation

Stochastic Stability Analysis of Coupled Viscoelastic Systems with Nonviscously Damping Driven by White Noise

Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

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