二阶随机参激系统的Lyapunov指数和稳定性

Maximal Lyapunov Exponent and Almost-Sure Sample Stabilityfor Second-Order Stochastic System

研究了二阶系统在随机噪声激励下的主共振响应和稳定性问题。用多尺度法确定了系统响应的幅值和相位角所满足的方程 ,讨论了系统的阻尼项、随机项等对系统响应的影响。求出了系统的不变测度和最大 L yapunov指数的解析表达式 ,由最大 L yapunov指数可得系统几乎必然稳定的充分必要条件。

The principal resonance and stability of second-order stochastic system to random parametric excitation were investigated. The method of multiple scales was used to determine the equations of the modulation of amplitude and phase. The effects of damping, detuning, bandwidth, and magnitudes of random excitation are analyzed. The explicit asymptotic formulas for the maximum Lyapunov exponent are obtained. The almost sure stability or instability of the stochastic Mathieu system depends on the sign of the maximum Lyapunov exponent.