Name: 贺一珺
Student Number: 2014302290002

Question

• 3.30 Investigate the Lyapunov exponent of the stadium billiard for several values of \alpha. You can do this qualitatively by examing the behavior for only one set of initial conditions for each value of \alpha you consider, or more quantitatively by averaging over a range of initial conditions for each value of \alpha.

• 3.31 Study the behavior for other types of tables. One interesting possibility is a square table with a circular interior wall located either in the center, or slightly off-center. Another possibility is an elliptical table.

Abstract

In last homework I investigated the process to chaos, using the bifurcation diagram, which is quite stimulating. Now I will get closer to the world of chaos. I will show you two totally different systems which will also lead you to the real world which is full of chaos. They are Lorenz model which comes from weather problems and the billiard problem which lays in our daily life. There are also some new tricks that you must pay attention to when doing your progremming and I will give you a brief introduction to those small tricks. Let's prolong our legs kicking tricks together! I hate tricks! Also, I will show you the trejectory and phase space of the billiard problem which in my opinion will certainly makes you upsurge!

Background

Lorenz system[1]

The Lorenz system is a system of ordinary differential equations (the Lorenz equations, note it is not Lorentz) first studied by Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system which, when plotted, resemble a butterfly or figure eight.

Intermittent_Lorenz_Attractor_-_Chaoscope

The model is a system of three ordinary differential equations now known as the Lorenz equations:

Here x, y, and z make up the system state, t is time, and σ, ρ, β are the system parameters. The Lorenz equations also arise in simplified models for lasers, dynamos, thermosyphons, brushless DC motors, electric circuits, chemical reactions and forward osmosis.

Billiard ball[2]

A billiard ball is a small, hard ball used in cue sports, such as carom billiards, pool, and snooker. The number, type, diameter, color, and pattern of the balls differ depending upon the specific game being played. Various particular ball properties such as hardness, friction coefficient and resilience are important to accuracy.

Billiards_balls

Crutial points

• The system gives you a path to chaos. Thus you have to minimize every possible error. When handeling the billiard problem, you have to take care of boundary carefully. Since we devided the total time into many small parts, the volue on the boundary are actually skipped. What should we do is to add code and approxe the boundary to ecrease the error.
• The hardest problem I meet in this homework is how to progrem the boundary condition correctly. The solotion could be approached by adjust the variables in the situation of simple circle.

Plotting

Before the begining

Lorenz Model

I tried out Lorenz model in our textbook. Here is initial condition:
Time step = 0.0001
σ = 10
b = 8/3
r = 25

Lorenz Model z versus time

3D trejectory

I adjusted the volue of r and plotted again. In this picture the volue of r are 25, 10, 5, retrospectively.

z versus time

It can be derived from this picture that when r = 25 the system is in the state of chaos actually.

The Billiard Problem

I also tired out some simple billiard problems in this section, which include a billiard on a square table and a billiard on a circle table.

Square table

Initial Condition:
Time step = 0.01
x = 0.2
y = 0
vx = 1
vy = 0.6

trajectory of a billiard on a table

phase space plot

Circular table

Initial condition:
x = 0.2
y = 0
vx = 1
vy = 0.6

trajectory of a billiard on a table

phase space

It's beautiful, isn't it? Now let's begin our work.

Problem 3.30

Plot the trajectory on four possible table first. Note that I used the trick mensioned before here. The initial condition is the same as last plot.

the trajectory on four possible table

phase space

Here is the seperation versus time when alpha is 0:

alpha = 0

Here is the seperation versus time when alpha is 0.001:

alpha = 0.001

Here is the seperation versus time when alpha is 0.01:

alpha = 0.01

Here is the seperation versus time when alpha is 0.1:

alpha = 0.1

Problem 3.31

Square table with a circular interior wall

trajectory of a billiard on a table

phase space

Elliptical table

trajectory of a billiard on a table

Phase space

A 3D version by vpython

Circular table

Conclution

I investigated two more systems in this homework, but there are still some problems. The first is that the phase space of 3.30 is actually wrong. However, it's hard for me to figure out the crutial points. Also, I got some stimulating results in this homework. These two sytems exhibit all charicters of chaos.

Acknowledgement

[1]wikipedia
[2]Shuxu Hu(胡墅绪) for that he helped me using vpython to work out 3D diagrem
[3]Shan Tan(谭善) for his idea on velocity calculating near boundary
[4]Prof. Cai

How to contact me

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