谐和与噪声联合作用下Duffing振子的安全盆分叉与混沌

Bifurcations of safe basins and chaos in softening Duffing oscillator under harmonic and bounded noise excitation

研究了软弹簧Duffing振子在确定性谐和外力和有界随机噪声联合作用下,系统安全盆的侵蚀和混沌现象.推导出系统的随机Melnikov过程,根据Melnikov过程在均方意义上出现简单零点的条件给出了系统出现混沌的临界值,然后用数值模拟方法计算了系统的安全盆分叉点.结果表明,由于随机扰动的影响,系统的安全盆分叉点发生了偏移,并且使得混沌容易发生.

The erosion of the safe basins and related chaotic motions of a softening Duffing oscillator under harmonic and bounded random noise are studied. By the Melnikov method, the system＇s Melnikov integral is computed and the parametric threshold for the onset of chaos is obtained. Using the Monte-Carlo and Runge-Kutta method, the erosion of safe basins is also discussed. As an alternative definition, stochastic bifurcation may be defined as a sudden change in the character of stochastic safe basins when the bifurcation parameter of the system passes through a critical value. This definition applies equally well to either randomly perturbed motions or purely deterministic motions. It is found that random noise may destroy the integrity of the safe basins, bringing forward the stochastic bifurcation and making the threshold for onset of chaos vary to a large extent, which makes the system less safe and chaotic motion easier to occur.