First-passage time of Duffing oscillator under combined harmonic and white-noise excitations

The first-passage time of Duffing oscillator under combined harmonic and white-noise excitations is studied. The equation of motion of the system is first reduced to a set of averaged Ito stochastic differential equations by using the stochastic averaging method. Then, a backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, and the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. Numerical results for two resonant cases with several sets of parameter values are obtained and the analytical results are verified by using those from digital simulation.

Ancient Chinese algorithm: The Ying Buzu Shu (method of surplus and deficiency) vs Newton iteration method

Air exploratory discussion of an ancient Chinese algorithm, the Ying Buzu Shu, in about 2nd century BC, known as the rule of double false position in the West is given. In addition to pointing out that the rule of double false position is actually a translation version of the ancient Chinese algorithm, a comparison with well-known Newton iteration method is also made. If derivative is introduced, the ancient Chinese algorithm reduces to the Newton method. A modification of the ancient Chinese algorithm is also proposed, and some of applications to nonlinear oscillators are illustrated.

二阶非线性微分方程亚/超谐共振的一种求解方法及其应用研究

本文引用了非线性共振频率的概念,指出如何求主共振频率域及其中心。采用新的办法对单自由度非线性系统的亚/超谐共振进行了研究。作为例子讨论了Duffing方程,给出了1/7,1/5,1/2,3/5,5/3,2,5次谐波共振的结果。模拟计算机的计算结果对所给理论作了部分验证。

The nonlinear evolution equation on the shallow water wave

Based on the variation principle, the nonlinear evolution model for the shallow water waves is established. The research shows the Duffing equation can be introduced to the evolution model of water wave with time.

非线性海洋波动的研究

本文围绕非线性海洋波动理论分别从三个方面作了研究。第一，非线性定形波的研究。采用微分方程的几何理论与动力学相结合的方法，将控制地球流体的运动方程，通过行波变换，作为平面自治系统，利用相图理论，分别分析了非线性惯性重力波、非线性Rossby 波等的定性性质，得到了惯性重力波不存在定形孤波解的结论；通过对平面自治系统在平衡点处作Taylor展开，结合K－B平均法，求得了带有刻划非线性效应小参数的非线性频散关；论证了分式简谐函数为地球流体中有限振幅波解的一般形式。第二，非线性水波时空变化规律的研究，利用水波的变分方程，在行波或空间上进行Galerkin截谱，分离时空变量，并计及非线性效应的二次项，分别导出了水波随时间、空间的变化规律。其中，浅水非笥性波随时、空的变化规律满足用来描述非线性振动现象的Duffing方程，这使得Duffing方程在非线性水波的研究领域也找到了应用背。众所财拓KdV理论是水波的非线性效应和色散效应平衡的产物，在此尺度下，波形随时、空的变化不能被揭示出来。因此，本文所得到的结果是KdV理论和调制波理论的补充和发展。而且，这些结果便于实际应用。第三，非线性随机海浪统计分布的研研。基于动力学原理，研究了非线性波振动和位相的分布规律。利用我们所得到的波形随时间的演化过程，将随机输入看作初值，用渐近方法得到带有随机项的非线性水波的振幅和位相，从而建立了符合动力学原理的非线性波振幅和位相的概率分布，将将其和传统的结论以及测资料作了比较，结果表明，吻合较好。这种尝试性的方法在该领域的应用尚属首次，可望能够成为研究非性线随机波的一种手段。