Moment Lyapunov Exponent for Two Coupled Oscillators Driven by Real Noise

Namachchivaya, N. Sri; Roessel, H. J. Van; Doyle, Monica M.

doi:10.1137/s003613999528138x

Cited by 33 publications

(15 citation statements)

References 17 publications

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“…. R7m,+l (~rn+l> Therefore from (7) and (19) Consequently substituting this estimate into (26) and using Gronwall inequality again we finally obtain (17).…”

Section: Downloaded By [Tobb Ekonomimentioning

confidence: 82%

“…Recently Kao and Wihstutz in [16] studied almost sure stabilization of companion form systems by means of mean zero degenerate real noise and showed that the damped inverse pendulum can be almost sure stabilized by such noise. It is known that in applications the mean square stability is maybe more adequate than almost sure stability (see, e.g., a discussion in [26] and [4]). This is the reason to study in this paper the mean square stabilization of a time-varying deterministic linear system .…”

Section: Introductionmentioning

confidence: 99%

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Mean square stabilization of linear systems by mean zero noise

Bobryk

Stettner

1999

Stochastics and Stochastic Reports

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“…. R7m,+l (~rn+l> Therefore from (7) and (19) Consequently substituting this estimate into (26) and using Gronwall inequality again we finally obtain (17).…”

Section: Downloaded By [Tobb Ekonomimentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Mean square stabilization of linear systems by mean zero noise

Bobryk

Stettner

1999

Stochastics and Stochastic Reports

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show abstract

“…This research is motivated by problems in the dynamic stability of viscoelastic systems subjected to stochastically fluctuating loads. The paper is different from [12], where a perturbation approach was used, and is unlike [4], where the Girsanov theorem and Feynman-Kac formula were applied and the viscoelasticity was not considered. The study in this paper is an extension of [13] from white noise process to real noise process, that is, Ornstein-Uhlenbeck process.…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

“…The solution of the set of equations (12) converges in the weak sense and up to first order in to a diffusive Markov process whose governing Itô equations are of the form …”

Section: Formulationmentioning

confidence: 99%

Stochastic Stability of Coupled Viscoelastic Systems Excited by Real Noise

Deng

2018

Mathematical Problems in Engineering

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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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“…The small noise expansions of the moment Lyapunov exponent and the Lyapunov exponent for this two degree of freedom system are obtained in this paper. We consider a stochastic excitation with speci® c in® nitesimal generator G as in Sri Namachchivaya et al (1996). We derive the generator L …p †, whose principal eigenvalue is the moment Lyapunov exponent using a method based on an asymptotic expansion similar to that presented in Sri Namachchivaya et al (1996).…”

Section: …3 †mentioning

confidence: 99%

Stabilization of linear systems by noise: Application to flow induced oscillations

Namachchivaya

Vedula

2000

Dynamics and Stability of Systems

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In a recent paper, Popp and Romberg reported on stabilization by grid generated turbulence of a smooth circular cylinder immersed in the wake from an identical cylinder in an array of aluminium tubes. Although these results were obtained experimentally, so far they have not been explained from a stochastic point of view on a rigorous theoretical basis. This paper provides analytical results which may explain this stabilization phenomenon by modelling the immersed cylinder as a two degree of freedom oscillator and the turbulence as a stochastic process. We obtain general asymptotic approximation for the moment Lyapunov exponent, g…p †, and the Lyapunov exponent, ¶, for a four-dimensional system with one critical mode and another asymptotically stable mode driven by a small intensity stochastic process. These results, pertaining to pth moment stability and almost-sure stability, explain how the stochastic components that couple the stable and the critical modes play an important role in determining whether a noisy excitation can stabilize or destabilize the oscillatory critical mode. They are then applied to a prototypical¯ow induced oscillation model to justify the experimental results.

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Moment Lyapunov Exponent for Two Coupled Oscillators Driven by Real Noise

Cited by 33 publications

References 17 publications

Mean square stabilization of linear systems by mean zero noise

Mean square stabilization of linear systems by mean zero noise

Stochastic Stability of Coupled Viscoelastic Systems Excited by Real Noise

Stabilization of linear systems by noise: Application to flow induced oscillations

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