【摘要】 研究了软弹簧Duffing振子在多频率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。将Melnikov方法推广到包含有限多个频率外力和随机噪声联合作用的情形,推导出了系统的随机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。同时证明:激励频率数目的增加使得系统产生混沌的参数临界值变小,也使得安全盆分叉提前发生,系统变得不安全。更多还原

【Abstract】 The erosion of the safe basins and related chaotic motions of a softening Duffing oscillator under multi-frequency external periodic forces and bounded random noise are investigated.The system Melnikov integral and the parametric threshold for onset of chaos are obtained.The Melnikov global perturbation technique is therefore generalized to higher dimensional systems,and the erosion of safe basins is 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,which applies successfully to either randomly perturbed motions,or purely deterministic motions.It is found that increasing the number of forcing frequencies or increasing the random noise may destroy the integrity of the safe basins to lead to the occurrence of the stochastic bifurcation and make the threshold for onset of chaos vary in a larger extent.更多还原