Moment Lyapunov Exponent for Two Coupled Oscillators Driven by Real Noise

1996 ◽  
Vol 56 (5) ◽  
pp. 1400-1423 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
H. J. Van Rossel ◽  
M. M. Doyle
Keyword(s):  
Lyapunov Exponent ◽  
Coupled Oscillators ◽  
Real Noise ◽  
Moment Lyapunov Exponent

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.

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).

SYMMETRY BREAKING BIFURCATIONS OF CHAOTIC ATTRACTORS

1995 ◽  
Vol 05 (06) ◽  
pp. 1643-1676 ◽  
Author(s):  
PHILIP J. ASTON ◽  
MICHAEL DELLNITZ
Keyword(s):  
Lyapunov Exponent ◽  
Symmetry Breaking ◽  
Coupled Oscillators ◽  
Dynamical Behavior ◽  
Invariant Sets ◽  
Chaotic Attractors ◽  
Loss Of Stability ◽  
Efficient Computation ◽  
The Stability

In an array of coupled oscillators, synchronous chaos may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classification of Lyapunov exponents based on the symmetry of the underlying problem. This classification is crucial in the derivation of the analytical results but it also allows an efficient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries, and this leads to the idea of symmetry breaking bifurcations of chaotic attractors. Finally, the results and ideas are illustrated for several systems of coupled oscillators.

Sign in / Sign up

Export Citation Format

Share Document