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 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 a Planner Gyropendulum System with Synchronous Motor Driven by Gaussian White Noises

2019 ◽  
Vol 19 (10) ◽  
pp. 1971006
Author(s):  
Shenghong Li ◽  
Zurun Xu
Keyword(s):  
Lyapunov Exponent ◽  
Synchronous Motor ◽  
Infinite Matrix ◽  
Almost Sure Stability ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Moment Lyapunov Exponent ◽  
The Stability ◽  
The Moment ◽  
White Noises

In this paper, the stochastic moment stability and almost-sure stability of a planner gyropendulum system with synchronous motor under the white noises are investigated. By applying the theory of diffusion process, an eigenvalue problem for the moment Lyapunov exponent is formulated. Then, through a perturbation method and a Fourier cosine series expansion, the second-order expansion of the moment Lyapunov exponent is solved, which is just the leading eigenvalue of an infinite matrix. Finally, the convergence and validity of the procedure are numerically verified, and the effects of system and noise parameters on the moment Lyapunov exponent are discussed. It was found that the increase in both the noise intensity and coefficient of the synchronous motor torque will weaken the stability of the gyropendulum system, and when they reach certain values, the system becomes unstable. In addition, according to the relationship between the moment Lyapunov exponent and maximal Lyapunov exponent, the stable thresholds are also given.

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.

MOMENT LYAPUNOV EXPONENT FOR CONSERVATIVE SYSTEMS WITH SMALL PERIODIC AND RANDOM PERTURBATIONS

2002 ◽  
Vol 02 (01) ◽  
pp. 25-48 ◽  
Author(s):  
P. IMKELLER ◽  
G. N. MILSTEIN
Keyword(s):  
Lyapunov Exponent ◽  
Autonomous Systems ◽  
Stability Index ◽  
Stationary Points ◽  
First Case ◽  
Conservative Systems ◽  
Moment Lyapunov Exponent ◽  
The Stability ◽  
The Moment ◽  
The Hill

Much effort has been devoted to the stability analysis of stationary points for linear autonomous systems of stochastic differential equations. Here we introduce the notions of Lyapunov exponent, moment Lyapunov exponent, and stability index for linear nonautonomous systems with periodic coefficients. Most extensively we study these problems for second order conservative systems with small random and periodic excitations. With respect to relations between the intrinsic period of the system and the period of perturbations we consider the incommensurable and commensurable cases. In the first case we obtain an asymptotic expansion of the moment Lyapunov exponent. In the second case we obtain a finite expansion except in situations of resonance. As an application we consider the Hill and Mathieu equations with random excitations.

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.

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.

scholarly journals Stochastic Stability of Coupled Viscoelastic Systems Excited by Real Noise

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Jian Deng
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Stochastic Stability ◽  
Random Fluctuation ◽  
Largest Lyapunov Exponent ◽  
Real Noise ◽  
The Largest Lyapunov Exponent ◽  
The Stability ◽  
The Moment ◽  
Moment Lyapunov Exponents

The moment stochastic stability and almost-sure stochastic stability of two-degree-of-freedom coupled viscoelastic systems, under the parametric excitation of a real noise, are investigated through the moment Lyapunov exponents and the largest Lyapunov exponent, respectively. The real noise is also called the Ornstein-Uhlenbeck stochastic process. For small damping and weak random fluctuation, the moment Lyapunov exponents are determined approximately by using the method of stochastic averaging and a formulated eigenvalue problem. The largest Lyapunov exponent is calculated through its relation with moment Lyapunov exponents. The stability index, the stability boundaries, and the critical excitation are obtained analytically. The effects of various parameters on the stochastic stability of the system are then discussed in detail. Monte Carlo simulation is carried out to verify the approximate results of moment Lyapunov exponents. As an application example, the stochastic stability of a flexural-torsional viscoelastic beam is studied.

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