The moment Lyapunov exponent for a three-dimensional stochastic system

2014 ◽  
Vol 68 ◽  
pp. 40-47
Author(s):  
Xuan Li ◽  
Xianbin Liu
Keyword(s):  
Lyapunov Exponent ◽  
Stochastic System ◽  
Three Dimensional ◽  
Moment Lyapunov Exponent ◽  
The Moment

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.

THE ASYMPTOTIC BEHAVIOR OF SEMI-INVARIANTS FOR LINEAR STOCHASTIC SYSTEMS

2002 ◽  
Vol 02 (02) ◽  
pp. 281-294
Author(s):  
G. N. MILSTEIN
Keyword(s):  
Asymptotic Behavior ◽  
Lyapunov Exponent ◽  
Linear System ◽  
Characteristic Function ◽  
Stochastic Systems ◽  
Random Variable ◽  
Characteristic Functions ◽  
Linear Stochastic Systems ◽  
Moment Lyapunov Exponent ◽  
The Moment

The asymptotic behavior of semi-invariants of the random variable ln |X(t,x)|, where X(t,x) is a solution of a linear system of stochastic differential equations, is connected with the moment Lyapunov exponent g(p). Namely, it is obtained that the nth semi-invariant is asymptotically proportional to the time t with the coefficient of proportionality g(n)(0). The proof is based on the concept of analytic characteristic functions. It is also shown that the asymptotic behavior of the analytic characteristic function of ln |X(t,x)| in a neighborhood of the origin of the complex plane is controlled by the extension g(iz) of g(p).

Simulation of Moment Lyapunov Exponents for Linear Homogeneous Stochastic Systems

2009 ◽  
Vol 76 (3) ◽  
Author(s):  
Wei-Chau Xie ◽  
Qinghua Huang
Keyword(s):  
Lyapunov Exponents ◽  
Stochastic System ◽  
Stochastic Systems ◽  
Diffusion Processes ◽  
Numerical Algorithms ◽  
Stochastic Dynamical Systems ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Moment Lyapunov Exponents

Moment Lyapunov exponents are important characteristic numbers for describing the dynamic stability of a stochastic system. When the pth moment Lyapunov exponent is negative, the pth moment of the solution of the stochastic system is stable. Monte Carlo simulation approaches complement approximate analytical methods in the determination of moment Lyapunov exponents and provides criteria on assessing the accuracy of approximate analytical results. For stochastic dynamical systems described by Itô stochastic differential equations, the solutions are diffusion processes and their variances may increase with time. Due to the large variances of the solutions and round-off errors, bias errors in the simulation of moment Lyapunov exponents are significant in improper numerical algorithms. An improved algorithm for simulating the moment Lyapunov exponents of linear homogeneous stochastic systems is presented in this paper.

Moment Lyapunov Exponent and Stochastic Stability of Two Coupled Oscillators Driven by Real Noise

2001 ◽  
Vol 68 (6) ◽  
pp. 903-914 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
H. J. Van Roessel
Keyword(s):  
Lyapunov Exponent ◽  
Asymptotic Approximation ◽  
Coupled Oscillators ◽  
Stability Index ◽  
Stationary Solutions ◽  
Real Noise ◽  
Moment Lyapunov Exponent ◽  
The Stability ◽  
The Moment

In a recent paper an asymptotic approximation for the moment Lyapunov exponent, gp, of two coupled oscillators driven by a small intensity real noise was obtained. The utility of that result is limited by the fact that it was obtained under the assumption that the moment p is small, a limitation which precludes, for example, the determination of the stability index. In this paper that limitation is removed and an asymptotic approximation valid for arbitrary p is obtained. The results are applied to study the moment stability of the stationary solutions of structural and mechanical systems subjected to stochastic excitation.

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 DYNAMIC BEHAVIOR OF TWO ELASTICALLY CONNECTED NANOBEAMS UNDER A WHITE NOISE PROCESS

2020 ◽  
Vol 18 (2) ◽  
pp. 219
Author(s):  
Ivan R. Pavlović ◽  
Ratko Pavlović ◽  
Goran Janevski ◽  
Nikola Despenić ◽  
Vladimir Pajković
Keyword(s):  
Lyapunov Exponent ◽  
White Noise ◽  
Axial Loading ◽  
Regular Perturbation ◽  
White Noise Process ◽  
Noise Process ◽  
Asymptotic Expressions ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Moment Stability

This paper investigates the almost-sure and moment stability of a double nanobeam system under stochastic compressive axial loading. By means of the Lyapunov exponent and the moment Lyapunov exponent method the stochastic stability of the nano system is analyzed for different system parameters under an axial load modeled as a wideband white noise process. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises.

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