一类含噪非线性自参系统单模解的几乎必然稳定性
Almost-Sure Stability of the Single Mode Solution of a Noisy Nonlinear Autoparametric System
摘要：
对于悬挂在受白噪声激励的振动块下方的单摆，其保持垂直的解称为单模解。当这种解决方案变得不稳定时，能量从块转移到摆锤，有助于吸收块的振动。我们研究了控制沿单模解线性化的过程的几乎必然稳定性的李雅普诺夫指数。线性化方程由白噪声和色噪声过程的组合激励，这使得李雅普诺夫指数的评估非平凡的。当白噪声强迫强度趋于零时，我们得到了李雅普诺夫指数的三阶渐近表达式。
英文摘要：
For a pendulum suspended below a vibrating block with white noise forcing, the solution in which the pendulum remains vertical is called the single mode solution. When this solution becomes unstable there is energy transfer from the block to the pendulum, helping to absorb the vibrations of the block. We study the Lyapunov exponent governing the almost-sure stability of the process linearized along the single mode solution. The linearized equation is excited by a combination of white and colored noise processes, which makes the evaluation of the Lyapunov exponent non trivial. We obtain an explicit third order asymptotic expression for the Lyapunov exponent as the intensity of the white noise forcing tends to zero.
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