Maximal Lyapunov Exponent and Almost-Sure Stability for Coupled Two-Degree-of-Freedom Stochastic Systems

1994 ◽  
Vol 61 (2) ◽  
pp. 446-452 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
H. J. Van Roessel ◽  
S. Talwar
Keyword(s):  
Lyapunov Exponent ◽  
Stochastic Systems ◽  
Stochastic Averaging ◽  
Degree Of Freedom ◽  
Asymptotic Formulas ◽  
Perturbation Approach ◽  
Maximum Lyapunov Exponent ◽  
Almost Sure Stability ◽  
Asymptotic Growth Rate ◽  
Two Degree Of Freedom

In this paper, a perturbation approach is used to calculate the asymptotic growth rate of stochastically coupled two-degree-of-freedom systems. The noise is assumed to be white and of small intensity in order to calculate the explicit asymptotic formulas for the maximum Lyapunov exponent, The Lyapunov exponents and rotation number for each degree-of-freedom are obtained in the Appendix. The almost-sure stability or instability of the four-dimensional stochastic system depends on the sign of the maximum Lyapunov exponent. The results presented here match those presented by the first author and others using the method of stochastic averaging, where approximate Itoˆ equations in amplitudes and phase are obtained in the sense of weak convergence.

Stochastic Stability of Linear Gyroscopic Systems: Application to Pipes Conveying Fluid

2002 ◽  
Author(s):  
Lalit Vedula ◽  
N. Sri Namachchivaya
Keyword(s):  
Lyapunov Exponent ◽  
Stochastic Stability ◽  
Perturbation Approach ◽  
Almost Sure Stability ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Gyroscopic Systems ◽  
Two Degree Of Freedom ◽  
Pulsating Fluid ◽  
Moment Stability

In this paper we obtain asymptotic approximations for the moment Lyapunov exponent, g(p), and the Lyapunov exponent,λ, for a two-degree-of-freedom gyroscopic system close to a double zero resonance and subjected to small damping and noisy disturbances. Using a perturbation approach, we show analytically that the moment and the top Lyapunov exponent grow in proportion to ε1/3 when the damping and noise respectively are of O(ε) and O(ε). These results, pertaining to pth moment stability and almost-sure stability of the trivial solution, are applied to study the stochastic stability of a pipe conveying pulsating fluid.

Stochastic Stability of Coupled Oscillators in Resonance: A Perturbation Approach

2004 ◽  
Vol 71 (6) ◽  
pp. 759-768 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
H. J. Van Roessel
Keyword(s):  
Lyapunov Exponent ◽  
Eigenvalue Problem ◽  
Coupled Oscillators ◽  
Orthogonal Expansion ◽  
Perturbation Approach ◽  
Almost Sure Stability ◽  
Moment Lyapunov Exponent ◽  
The Stability ◽  
The Moment ◽  
The Galerkin Method

A perturbation approach is used to obtain an approximation for the moment Lyapunov exponent of two coupled oscillators with commensurable frequencies driven by a small intensity real noise with dissipation. The generator for the eigenvalue problem associated with the moment Lyapunov exponent is derived without any restriction on the size of pth moment. An orthogonal expansion for the eigenvalue problem based on the Galerkin method is used to derive the stability results in terms of spectral densities. These results can be applied to study the moment and almost-sure stability of structural and mechanical systems subjected to stochastic excitation.

scholarly journals Higher order stochastic averaging method for mechanical systems having two degrees of freedom

2004 ◽  
Vol 26 (2) ◽  
pp. 103-110
Author(s):  
Nguyen Duc Tinh
Keyword(s):  
Degrees Of Freedom ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Higher Order ◽  
Degree Of Freedom ◽  
Approximation Solution ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Two Degree Of Freedom

Higher order stochastic averaging method is widely used for investigating single-degree-of-freedom nonlinear systems subjected to white and coloured random noises.In this paper the method is further developed for two-degree-of-freedom systems. An application to a system with cubic damping is considered and the second approximation solution to the Fokker-Planck (FP) equation is obtained.

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