Name: 贺一珺
Student Number: 2014302290002

Question

• 3.12 In constructing the Poincare section we plotted only at times that were in phase with the drive forse; that is, at times t=2npi/frequency of force, where n is an integer. At these values of t the driving force passed through zero. However, we could just as easily have chosen to make the plot at times corresponding to a maximum of the drive force, or at times pi/4 out-of-phase with this force, etc. Construct the Poincare sections for these cases and compare them with Figure 3.9.

• 3.13 Write a progrem to calculate and compare the behavior of two, nearly identical pendulums. Use it to calculate the divergence of two nearby trajectorys in the chaotic regime, and make a qualitative estimate of the corresponding Lyapunov exponent from the slope of a plot of log\theta as a function of t.

• 3.14 Repeat the previous problem, but give the two pendulums slightly different damping factors. How does the value of the Lyapunov exponent compare with that found in Fighre 3.7?

Abstract

It is easy to list the nonlinear diffrential equation of this question. But it is hard to get the exact solution analytically. Thus I use Euler-Cromer method instead of Euler method which we learned in previous chapter to solve this equation numberically. I design a progrem which allow one to put in initial conditions including initial angle, initial angular speed, friction coefficient, the magnificance of force and the frequency of force. It can be used to calculate the angle and the difference of angle of two systems with different initial conditions. It is quite a bit interesting to observe the motion of pendulums under different initial conditions. The chaos can also be observed using this programming.
Also, in the phase-space graph,I will plot angle versus angular velocity only at times that are in phase with the driving force which is powerful when analyse the system.

Background

• Pendulum

  
  EastHanSeismograph
  

  A pendulum is a weight suspended from a pivot so that it can swing freely. One of the earliest known uses of a pendulum was a 1st-century seismometer device of Han Dynasty Chinese scientist Zhang Heng. The equations of motion of a pendulum can be expressed as:

In this project, Euler-Cromer Method is used. Thus it can be deduced that:

• Chaos theory
  From wikipedia we know that Chaos theory is the field of study in mathematics that studies the behavior of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos.
  What's more, chaos is also a traditional Chinese term which means the most original state which has no rules and laws. In Chanese mythology, it was Pangu who broke the state of chaos of our universuty, and then the laws of the nature came out "naturely". It's certainly interesting with deep thoughts of metaphysics in it, but it's out of our course.

  
  盘古

Realization

1. Creat the class of pendulums and set the initial conditions of the system.
2. Creat several lists where you can save your datas in.
3. Define the function to do the calculation and plotting.
4. Rearrange the progrem so that you can input the initial conditions and the progrem could work out the angle versus time automatically.

Plotting

The gate to the question

Clike here to get code
In this section I want to show you some pictures of forced vibration with damping factor under different magnitude of force and of the state of chaos to introduce you in to the wonderful world of viberation.
I set the damping factor to be 0.5 and the initial angle to be 0.2rad. Also, the frequency of force is 2/3 s^-1. Mgnitude of force are shown in picture:

受迫振动

It can be easily observed that the state of chaos appears in the 3rd line when the magnitude of force is set to be 1.2N.
Details are showed in the following picture:

混沌

Since we have gain the knowledge of the basic graphs of forced viberance in former pictures, let's begin to solve the questions ^_

Solutions for questions

• 3.12
  Clike here to get code
  I set the damping factor to be 0.5 and initial angle to be 0.2 rad. Also, the magnitude of force is 1.2N and the frequency of force is 2/3 s^-1.
  The time condition has been marked on the picture.
  The picture is as following:
  phase space
  And then, I set four different time conditions and ploted them in the following picture:

phase space

Clike here to get the code of the following solutions

• 3.13
  To get the chaotic regime, we set the damping factor to be 0.5 ,the magnitude of force to be 1.2N and the frequency of force to be 2/3 s^-1.
  I plotted the picture which set initial angle to be 0.0000rad and 0.0001rad respectively.
  The picture of angle versus time is as following:

Angle

The difference of angle under two initial conditions versus time was plotted also:

Difference of angle

• 3.14
  I modified the damping conditions in the last question slightly and plotted the angle under different damping condition:
  Modify the damping factor to q=0.49:
  The angle versus time is:

  
  Angle
  

  The difference of angle is:

Difference of angle

Modify the damping factor to q=0.51:
The angle versus time is:

Angle

The difference of angle is:

Difference of angle

To work deeper, I also do the contrust of the same initial angle(0rad) under different damping factors, here is the results:

The picture of angle versus time is:

Angle

The difference of angle when q1=0.5 and q2=0.49 is:

Difference of angle

The difference of angle when q1=0.5 and q2=0.49 is:

Difference of angle

Acknowledge

• Prof. Cai
• Wikipedia

How to contact me

• Wechat ID: bestsola
• Phone number: 18827628190
• E-mail: 2014302290002@whu.edu.cn