Weak Noise Expansion of Moment Lyapunov Exponents of a Two-Dimensional System Under Bounded Noise Excitation

2000 ◽  
Author(s):  
Wei-Chau Xie
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Dimensional System ◽  
Two Dimensional ◽  
Bounded Noise ◽  
Regular Perturbation ◽  
Weak Noise ◽  
Two Dimensional System ◽  
The Moment ◽  
Moment Lyapunov Exponents

Abstract The moment Lyapunov exponents of a two-dimensional system under bounded noise parametric excitation are studied in this paper. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter.

Moment Lyapunov Exponents of a Two-Dimensional Viscoelastic System Under Bounded Noise Excitation

2002 ◽  
Vol 69 (3) ◽  
pp. 346-357 ◽  
Author(s):  
W.-C. Xie
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Two Dimensional ◽  
Bounded Noise ◽  
Regular Perturbation ◽  
Stochastic Fluctuation ◽  
Viscoelastic System ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Moment Lyapunov Exponents

The moment Lyapunov exponents of a two-dimensional viscoelastic system under bounded noise excitation are studied in this paper. An example of this system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The stochastic parametric excitation is modeled as a bounded noise process, which is a realistic model of stochastic fluctuation in engineering applications. The moment Lyapunov exponent of the system is given by the eigenvalue of an eigenvalue problem. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter. The results obtained are compared with those for which the effect of viscoelasticity is not considered.

scholarly journals MOMENT LYAPUNOV EXPONENTS AND STOCHASTIC STABILITY OF A THIN-WALLED BEAM SUBJECTED TO AXIAL LOADS AND END MOMENTS

2021 ◽  
Vol 19 (2) ◽  
pp. 209
Author(s):  
Goran Janevski ◽  
Predrag Kozić ◽  
Ratko Pavlović ◽  
Strain Posavljak
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Degrees Of Freedom ◽  
Simulation Method ◽  
Stochastic Dynamic ◽  
Regular Perturbation ◽  
Thin Walled ◽  
The Moment ◽  
Moment Stability ◽  
Moment Lyapunov Exponents

In this paper, the Lyapunov exponent and moment Lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation are investigated. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises. The Lyapunov exponent and moment Lyapunov exponents are important characteristics for determining both the almost-sure and the moment stability of a stochastic dynamic system. As an example, we study the almost-sure and moment stability of a thin-walled beam subjected to stochastic axial load and stochastically fluctuating end moments.  The validity of the approximate results for moment Lyapunov exponents is checked by numerical Monte Carlo simulation method for this stochastic system.

Numerical Determination of Moment Lyapunov Exponents of Two-Dimensional Systems

2005 ◽  
Vol 73 (1) ◽  
pp. 120-127 ◽  
Author(s):  
Wei-Chau Xie ◽  
Ronald M. C. So
Keyword(s):  
Lyapunov Exponents ◽  
Eigenvalue Problems ◽  
Principal Eigenvalue ◽  
Dimensional System ◽  
Two Dimensional ◽  
Bounded Noise ◽  
Partial Differential ◽  
Stochastic Dynamical System ◽  
Moment Lyapunov Exponents

The pth moment Lyapunov exponent of an n-dimensional linear stochastic system is the principal eigenvalue of a second-order partial differential eigenvalue problem, which can be established using the theory of stochastic dynamical system. An analytical-numerical approach for the determination of the pth moment Lyapunov exponents, for all values of p, is presented. The approach is illustrated through a two-dimensional system under bounded noise or real noise parametric excitation. Series expansions of the eigenfunctions using orthogonal functions are employed to transform the partial differential eigenvalue problems to linear algebraic eigenvalue problems, which are then solved numerically. The numerical values obtained are compared with approximate analytical results with weak noise amplitudes.

Numerical Determination of the Moment Lyapunov Exponents

2003 ◽  
Author(s):  
Wei-Chau Xie ◽  
Ronald M. C. So
Keyword(s):  
Lyapunov Exponents ◽  
Eigenvalue Problems ◽  
Double Series ◽  
Numerical Approach ◽  
Dimensional System ◽  
Bounded Noise ◽  
Partial Differential ◽  
The Moment ◽  
Moment Lyapunov Exponents

Two numerical methods for the determination of the pth moment Lyapunov exponents of a two-dimensional system under bounded noise or real noise parametric excitation are presented. The first method is an analytical-numerical approach, in which the partial differential eigenvalue problems governing the moment Lyapunov exponents are established using the theory of stochastic dynamical systems. The eigenfunctions are expanded in double series to transform the partial differential eigenvalue problems to linear algebraic eigenvalue problems, which are then solved numerically. The second method is a Monte Carlo simulation approach. The numerical values obtained are compared with approximate analytical results with weak noise amplitudes.

Lyapunov Exponents and Moment Lyapunov Exponents of a Two-Dimensional Near-Nilpotent System

2000 ◽  
Vol 68 (3) ◽  
pp. 453-461 ◽  
Author(s):  
W.-C. Xie
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Stochastic Perturbation ◽  
Pitchfork Bifurcation ◽  
Two Dimensional ◽  
Point Boundary ◽  
Moment Lyapunov Exponent ◽  
Linearized System ◽  
The Moment ◽  
Moment Lyapunov Exponents

The Lyapunov exponents and moment Lyapunov exponents of a near-nilpotent system under stochastic parametric excitation are studied. The system considered is the linearized system of a two-dimensional nonlinear system exhibiting a pitchfork bifurcation. The effect of stochastic perturbation in the vicinity of static pitchfork bifurcation is investigated. Approximate analytical results of Lyapunov exponent are obtained. The eigenvalue problem for the moment Lyapunov exponent is converted to a two-point boundary value problem, which is solved numerically by the method of relaxation.

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