Stochastic bifurcation in single-species model induced by α-stable Lévy noise

We study stochastic bifurcation for a system under multiplicative stable Lévy noise (an important class of non-Gaussian noise), by examining the qualitative changes of equilibrium states with its most probable phase portraits. We have found some peculiar bifurcation phenomena in contrast to the deterministic counterpart: (i) When the non-Gaussianity parameter in Lévy noise varies, there is either one, two or no backward pitchfork type bifurcations; (ii) When a parameter in the vector field varies, there are two or three forward pitchfork bifurcations; (iii) The non-Gaussian Lévy noise clearly leads to fundamentally more complex bifurcation scenarios, since in the special case of Gaussian noise, there is only one pitchfork bifurcation which is reminiscent of the deterministic situation.

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Stochastic bifurcation for two-time-scale dynamical system with α-stable Lévy noise

Journal of Statistical Mechanics Theory and Experiment ◽

10.1088/1742-5468/abdeb2 ◽

2021 ◽

Vol 2021 (3) ◽

pp. 033204

Author(s):

Shenglan Yuan ◽

Zhigang Zeng ◽

Jinqiao Duan

Keyword(s):

Dynamical System ◽

Time Scale ◽

Lévy Noise ◽

Stochastic Bifurcation ◽

Levy Noise

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Probability density function of leaky integrate-and-fire model with Levy noise and its numerical approximation

Сибирский журнал вычислительной математики ◽

10.15372/sjnm20160107 ◽

2016 ◽

Vol 19 (1) ◽

Keyword(s):

Probability Density Function ◽

Probability Density ◽

Density Function ◽

Numerical Approximation ◽

Lévy Noise ◽

Fire Model ◽

Integrate And Fire ◽

Levy Noise

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The stochastic Strichartz estimates and stochastic nonlinear Schrödinger equations driven by Lévy noise

Journal of Functional Analysis ◽

10.1016/j.jfa.2021.109021 ◽

2021 ◽

pp. 109021

Author(s):

Zdzisław Brzeźniak ◽

Wei Liu ◽

Jiahui Zhu

Keyword(s):

Nonlinear Schrödinger Equations ◽

Strichartz Estimates ◽

Lévy Noise ◽

Schrodinger Equations ◽

Schrödinger Equations ◽

Nonlinear Schrödinger ◽

Nonlinear Schrodinger Equations ◽

Levy Noise

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An Enhanced Intrinsic Time-Scale Decomposition Method Based on Adaptive Lévy Noise and Its Application in Bearing Fault Diagnosis

Symmetry ◽

10.3390/sym13040617 ◽

2021 ◽

Vol 13 (4) ◽

pp. 617

Author(s):

Jianpeng Ma ◽

Shi Zhuo ◽

Chengwei Li ◽

Liwei Zhan ◽

Guangzhu Zhang

Keyword(s):

Fault Diagnosis ◽

Time Scale ◽

Lévy Noise ◽

Model Parameters ◽

Rolling Bearings ◽

Intrinsic Time ◽

Time Scale Decomposition ◽

Weak Fault ◽

Levy Noise ◽

The Impact

When early failures in rolling bearings occur, we need to be able to extract weak fault characteristic frequencies under the influence of strong noise and then perform fault diagnosis. Therefore, a new method is proposed: complete ensemble intrinsic time-scale decomposition with adaptive Lévy noise (CEITDALN). This method solves the problem of the traditional complete ensemble intrinsic time-scale decomposition with adaptive noise (CEITDAN) method not being able to filter nonwhite noise in measured vibration signal noise. Therefore, in the method proposed in this paper, a noise model in the form of parameter-adjusted noise is used to replace traditional white noise. We used an optimization algorithm to adaptively adjust the model parameters, reducing the impact of nonwhite noise on the feature frequency extraction. The experimental results for the simulation and vibration signals of rolling bearings showed that the CEITDALN method could extract weak fault features more effectively than traditional methods.

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