摘要
研究了相关乘性和加性高斯白噪声激励下,双稳态Duffing-Van der Pol系统的随机P-分岔和D-分岔;利用随机平均法,得出系统幅值稳态概率密度的理论表达式,以及随机P-分岔发生的临界参数条件;通过分析概率密度曲线形状的变化,发现阻尼系数、加性和乘性噪声强度均可诱导系统出现随机P-分岔,但对系统分岔区域的影响有着明显的不同,同时Monte-Carlo数值模拟验证了理论分析的有效性.此外,利用Wolf算法得到系统的最大Lyapunov指数,并分析了系统的稳定性和随机D-分岔,发现加性和乘性噪声强度以及阻尼系数的增大,均会使系统趋于不稳定,而阻尼系数的增大,可以增强系统的稳定性.
关键词
随机分岔的研究始于20世纪80年代,引起了数学、物理、化学、工程等领域的普遍关注.随机分岔是非线性系统在噪声作用下产生的一种典型的非线性现象,反映了临界分岔系数对于弱随机扰动的敏感
一般情况下,加性噪声源于系统内部的涨落,而乘性噪声来源于系统所处外部环境的涨落,大多情况下乘性和加性噪声具有相互关联性,并可能对系统动力学产生较大的影响.曹力
考虑相关加性和乘性高斯白噪声激励下的Duffing-Van der Pol系统,其相应的随机微分方程可写为
(1) |
其中,,和分别是线性和非线性阻尼系数,为系统固有频率,是非线性刚度系数,且是一个小参量,取.和是相关的高斯白噪声,它们的统计性质满足以下条件:
(2) |
其中,和分别为加性噪声强度和乘性噪声强度,是乘性噪声和加性噪声之间的关联系数.
首先,对(1)式进行如下变
(3) |
利用(3)式将(1)式变成如下不相关的高斯白噪声激励下的系统:
(4) |
可以证明,和是不相关的高斯白噪声,的统计性质为
(5) |
这样,我们就把原方程(1)变成了由两个不相关的高斯白噪声激励下的系统(4).以下的分析计算都是针对(4)式进行的.
在确定性情形下,当时,系统(1)的相图中最多有两个吸引子共存:分别是在原点附近的平衡点和一个稳定的极限环.
为讨论噪声激励下系统(1)的分岔问题,以下应用随机平均法求解系统幅值的稳态概率密度函数.首先,引入如下变换:
(6) |
将(6)式代入(4)式,得到关于振幅和相位的标准方程为
(7) |
应用随机平均
(8) |
其中,是独立的单位Wiener过程,可以看出,振幅不依赖于的变化,故由(8)式可以得到关于振幅的Fokker-Planck-Kolmogorov (FPK)方程
(9) |
在(9)式中,令,可得稳态概率密度的表达式为
(10) |
其中,
(11) |
为归一化常数.
根据(10)式,利用拓扑性质的变化来讨论随机P-分岔.首先,根据可得以下代数方程:
(12) |
其中,
由(12)式可以得到系统的分岔参数临界条件为
(13) |
根据(13)式可以画出系统的三维分岔参数区间图,如

图1 系统(1)的三维分岔参数区间图
Fig.1 Three-dimensional bifurcation region diagram of system (1)

图2 时平面内的分岔区间图
Fig. 2 Bifurcation region diagram in the parameter plane when
由
在下面的分析中,我们讨论了不同参数对系统的稳态概率密度曲线的影响,并和数值模拟结果进行了对比.在Monte-Carlo数值模拟中,取初始条件为,,模拟数据长度.在

图3 时不同阻尼下振幅的稳态PDF曲线
Fig.3 Stationary PDF of amplitude under different damping coefficients when
下面我们研究了非线性阻尼系数和对系统分岔行为的影响.

图4 平面内的分岔区间图
Fig.4 Bifurcation region diagram in the parameter plane

图5 时不同阻尼下幅值的稳态PDF曲线
Fig.5 Stationary PDF of amplitude under different damping coefficients when
接下来,我们将分析噪声强度及噪声之间的关联系数对系统分岔行为的影响.

图6 不同D下平面内的分岔参数区间图
Fig.6 Bifurcation region diagram in the plane for different D

图7 不同下平面内的分岔参数区间图
Fig.7 Bifurcation region diagram in the plane for different
利用随机平均法计算稳态概率密度

图8 幅值稳态PDF随变化的曲线
Fig.8 Curves of stationary probability density function of amplitude for different
利用Oseledec乘法定
(14) |
其中,表征原方程解轨迹相邻两点间在时刻的距离,为欧几里德向量范数.通过(16)式可计算出系统的最大Lyapunov指数.当时,系统是不稳定的;当时,系统是稳定的.可以根据的符号变化判断系统随机D-分岔的发生.利用Wolf算
在

图9 时最大Lyapunov指数作为的函数随和阻尼系数变化的曲线
Fig.9 Largest Lyapunov exponent as a function of with different and when

图10 时最大Lyapunov指数作为的函数随变化的曲线
Fig.10 Largest Lyapunov exponent as a function of with different and

图11 时最大Lyapunov指数作为Q的函数随变化的曲线
Fig.11 Largest Lyapunov exponent as a function of multiplicative noise intensity with different and

图12 时最大Lyapunov指数作为Q的函数随变化的曲线
Fig.12 Largest Lyapunov exponent as a function of multiplicative noise intensity with different and
本文主要研究了相关乘性和加性高斯白噪声激励下双稳态Duffing-Van der Pol系统的随机分岔.利用随机平均法求解系统振幅的稳态概率密度,并用Monte-Carlo数值模拟验证了理论分析的有效性,同时利用Wolf算法计算得到了系统的最大Lyapunov指数,并对系统的稳定性作出分析.研究结果表明:系统阻尼系数、加性和乘性噪声强度对系统分岔参数区域的影响有着明显的不同,且均能诱导系统发生随机P-分岔;增大加性和乘性噪声强度,均会使系统由稳定趋于不稳定,并诱发随机D-分岔;增大阻尼系数和,可以增强系统的稳定性,而的增大会削弱系统的稳定性.改变噪声互关联系数对稳态概率密度的影响很小,但在较大强度的噪声激励下,噪声互关联系数对系统的稳定性影响较大.
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