Stochastic averaging of quasi integrable and resonant Hamiltonian systems excited by fractional Gaussian noise with Hurst index 1/2 < H < 1

2017 ◽  
Vol 30 (1) ◽  
pp. 11-19 ◽  
Author(s):  
Q.F. Lü ◽  
M.L. Deng ◽  
W.Q. Zhu
Keyword(s):  
Hamiltonian Systems ◽  
Gaussian Noise ◽  
Stochastic Averaging ◽  
Hurst Index ◽  
Fractional Gaussian Noise

Responses of Linear and Nonlinear Oscillators to Fractional Gaussian Noise With Hurst Index Between 1/2 and 1

2015 ◽  
Vol 82 (10) ◽  
Author(s):  
Mao Lin Deng ◽  
Wei Qiu Zhu
Keyword(s):  
Gaussian Noise ◽  
Nonlinear Oscillators ◽  
Equivalent Linearization ◽  
Hurst Index ◽  
Mean Square ◽  
Fractional Gaussian Noise ◽  
Weakly Nonlinear ◽  
Range Correlation ◽  
Linear And Nonlinear ◽  
Linear And Nonlinear Oscillators

The responses of linear and nonlinear oscillators to fractional Gaussian noise (fGn) are studied. First, some preliminary concepts and properties of fractional Brownian motion (fBm) and fGn with Hurst index 1/2<H<1 are introduced. Then, the exact sample solution, correlation function, spectral density, and mean-square value of the response of linear oscillator to fGn are obtained. Based on the sample solution, it is proved that the long-range correlation index of displacement response of linear oscillator is the same as that of excitation fGn, i.e., 2-2H, while the velocity response has no such long-range correlation. An interesting discovery is that the ratio of kinetic energy to total energy decreases as increasing Hurst index H. Finally, for the responses of one and two degrees-of-freedom (DOF) nonlinear oscillators to fGn, the equivalent linearization method is applied to obtain the sample functions, correlation functions and mean-square values of the responses. Plenty of digital simulation results are obtained to support these solutions. It is shown that the approximate solution is effective for weakly nonlinear oscillators and it is feasible to apply the equivalent linearization to study multi-DOF weakly nonlinear oscillators.

Stationary Response of Multidegree-of-Freedom Strongly Nonlinear Systems to Fractional Gaussian Noise

2017 ◽  
Vol 84 (10) ◽  
Author(s):  
Qiang Feng Lü ◽  
Mao Lin Deng ◽  
Wei Qiu Zhu
Keyword(s):  
Hamiltonian System ◽  
Gaussian Noise ◽  
Hurst Index ◽  
Fractional Gaussian Noise ◽  
Stationary Response ◽  
Completely Integrable ◽  
Strongly Nonlinear ◽  
Fractional Stochastic Differential Equations ◽  
Strongly Nonlinear System ◽  
Probability Densities

The stationary response of multidegree-of-freedom (MDOF) strongly nonlinear system to fractional Gaussian noise (FGN) with Hurst index 1/2 < H < 1 is studied. First, the system is modeled as FGN-excited and -dissipated Hamiltonian system. Based on the integrability and resonance of the associated Hamiltonian system, the system is divided into five classes: partially integrable and resonant, partially integrable and nonresonant, completely integrable and resonant, completely integrable and nonresonant, and nonintegrable. Then, the averaged fractional stochastic differential equations (SDEs) for five classes of quasi-Hamiltonian systems with lower dimension and involving only slowly varying processes are derived. Finally, the approximate stationary probability densities and other statistics of two example systems are obtained by numerical simulation of the averaged fractional SDEs to illustrate the application and compared with those from original systems to show the advantages of the proposed procedure.

Response of Quasi-Integrable and Resonant Hamiltonian Systems to Fractional Gaussian Noise

2021 ◽  
Vol 144 (1) ◽  
Author(s):  
Q. F. Lü ◽  
W. Q. Zhu ◽  
M. L. Deng
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Analytical Solutions ◽  
Gaussian Noise ◽  
Nonlinear Dynamical Systems ◽  
Original System ◽  
System Response ◽  
Fractional Gaussian Noise

Abstract The major difficulty in studying the response of multi-degrees-of-freedom (MDOF) nonlinear dynamical systems driven by fractional Gaussian noise (fGn) is that the system response is not Markov process diffusion and thus the diffusion process theory cannot be applied. Although the stochastic averaging method (SAM) for quasi Hamiltonian systems driven by fGn has been developed, the response of the averaged systems still needs to be predicted by using Monte Carlo simulation. Later, noticing that fGn has rather flat power spectral density (PSD) in certain frequency band, the SAM for MDOF quasi-integrable and nonresonant Hamiltonian system driven by wideband random process has been applied to investigate the response of quasi-integrable and nonresonant Hamiltonian systems driven by fGn. The analytical solution for the response of an example was obtained and well agrees with Monte Carlo simulation. In the present paper, the SAM for quasi-integrable and resonant Hamiltonian systems is applied to investigate the response of quasi-integrable and resonant Hamiltonian system driven by fGn. Due to the resonance, the theoretical procedure and calculation will be more complicated than the nonresonant case. For an example, some analytical solutions for stationary probability density function (PDF) and stationary statistics are obtained. The Monte Carlo simulation results of original system confirmed the effectiveness of the analytical solutions under certain condition.

scholarly journals Generalized Wavelet Fisher’s Information of1/fαSignals

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Julio Ramírez-Pacheco ◽  
Homero Toral-Cruz ◽  
Luis Rizo-Domínguez ◽  
Joaquin Cortez-Gonzalez
Keyword(s):  
Fisher Information ◽  
Gaussian Noise ◽  
Structural Breaks ◽  
Wavelet Domain ◽  
Fractional Gaussian Noise ◽  
Domain Definition ◽  
Potential Applications ◽  
Definition Of ◽  
The Time Domain ◽  
Fisher's Information

This paper defines the generalized wavelet Fisher information of parameterq. This information measure is obtained by generalizing the time-domain definition of Fisher’s information of Furuichi to the wavelet domain and allows to quantify smoothness and correlation, among other signals characteristics. Closed-form expressions of generalized wavelet Fisher information for1/fαsignals are determined and a detailed discussion of their properties, characteristics and their relationship with waveletq-Fisher information are given. Information planes of1/fsignals Fisher information are obtained and, based on these, potential applications are highlighted. Finally, generalized wavelet Fisher information is applied to the problem of detecting and locating weak structural breaks in stationary1/fsignals, particularly for fractional Gaussian noise series. It is shown that by using a joint Fisher/F-Statistic procedure, significant improvements in time and accuracy are achieved in comparison with the sole application of theF-statistic.

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