谐和与窄带随机噪声联合作用下Duffing系统的参数主共振

研究了Ｄｕｆｉｎｇ振子在谐和与窄带随机噪声联合激励下的参数主共振响应和稳定性问题．用多尺度法分离了系统的快变项，并求出了系统的最大Ｌｙａｐｕｎｏｖ指数．本文还分析了失稳及跳跃现象，及系统的阻尼项、非线性项、随机项、确定性参激强度对系统响应的影响．数值模拟表明本文提出的方法是有效的．

PRINCIPAL RESPONSE OF DUFFING OSCILLATOR TO COMBINED DETERMINISTIC AND NARROW BAND RANDOM PARAMETERIC EXCITATION 1)

The principal resonance of a Duffing oscillator to combined deterministic and narrow band random parametric excitations is investigated. In particular, the case in which the parametric terms sharing close frequencies is examined. For the first time, the method of multiple scales is used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response are studied by means of qualitative analyses. Jumps are shown to occur if the random excitation is small. The effects of damping, detuning, and magnitudes of deterministic and narrow band parametric excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that: 1. When the intensity of the random excitation increases, the trivial steady state solution may lose its stability and then the system may have a nontrivial steady state solution. 2. When the intensity of the random excitation increases, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. 3. Under some conditions the systems may have two steady state solutions. 4. Under some conditions the jumps may exist.