Exercise_07: Chapter 3 problem 3.13

一.摘要：

Write a progrem to calculate and compare the behavior of two,nearly identical pendulums.Use it to calculate the divergence of two nearby trajectories in the chaotic regime,as in Figure3.7,and make a qualitative estimate of the corresponding Lyapunov exponent from the slope of a plot of log($\Delta{\theta}$)as a function of t.

二.背景介绍(公式推导）：

1.The equation of the motion of pendulum: $$\ddot{\theta}=-\frac{g}{l}*sin{\theta}-q*{\dot\theta}+F_D*sin(\Omega_D*t)-\frac{g}{l}*sin{\theta}-q*{\dot\theta}+F_D*sin(\Omega_D*t),(3.1)$$

2.Two first-order differential equations: $$\dot{\omega}=-\frac{g}{l}*sin{\theta}-q*{\dot\theta}+F_D*sin(\Omega_D*t),(3.2)$$ $$\dot{\theta}=\omega,(3.3)$$

3.So,when $-\pi<=\theta<=\pi$: $$\omega_{i+1}=\omega_i+(-\frac{g}{l}*sin{\theta_i}-q*\omega_i+F_D*sin(\Omega_D*t_i)*\Delta{t},(3.4)$$ $$\theta_{i+1}=\theta_i+\omega_{i+1}*\Delta{t},(3.5)$$ $$when \theta<-\pi:\theta=\theta+2*\pi$$ $$when \theta>\pi:\theta=\theta-2*\pi$$

三.正文：

1.When $\Delta{t}=0.04s,F_D=0.5,0.7,0.8,0.9,1.0,1.1,1.2:$

2（1）.When $F_D=0.5,\Delta{t}=0.02s,0.04s:$

2（2）.When $F_D=1.2,\Delta{t}=0.02s,0.04s:$

3.When $F_D=1.1:$

4.代码

四.结论：

1.log$\Delta{\theta}$is proportional to $\lambda*t.$When 1.0<=F_D<=1.1,$\lambda>0$the pendulum begins to be in the chaotic regime;

2.From the picture2(When $F_D=0.5,\Delta{t}=0.02s,0.04s$),we can know that the results are convinced.

五.致谢：

Thanks to Jiajia Luo for referring to her program.