Exercise_09:The Billiard Problem

Abstract

This time there are examples about chaos in our daily life. We can find chaos in billiard problem. Compared with the earlier exercise about Poincare section and bifurcation diagram, this is a more complicated system.

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Background

Consider the problem of a ball moving without friction on a horizontal table. We imagine that there are walls at the edges of table that reflect the ball perfectly and that there is no frictional force between the ball and the table.The Lorenz equations can be written as:

Except for the collisions with the walls, the motion of the billiard is quite simple. Before collisions the velocity is constant so we have:

we can solve this by using Euler method.However, as for the process of collisions, we have the following equations:

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The Main body

Lorenz problem

Using Lorenz equations above as well as 3D plot, we can see that it gives some hints of an underlying regularity.

click here to see the code (with a time step of 0.0001 and the initial condition x=1,y=z=0)

This figure place us in the chaotic regime.We can see that even though the behavior is strongly chaotic, there is very high degree of regularity in the phase-space trajectory.

click here to see the code (with a time step of 0.0001)

Problem 3.30

Investigate the Lyapunov exponent of the stadium billiard for several values of alpha. You can do this qualitatively by examining 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.

we place the collipse motion of a billiard on a circle,the trajectory are figured:

We can see a regular and delicate pattern without chaos.click here to see the code

However, if we cut the table along the x axis, and pull the two semicircular halves apart (along y), a distance 2 alpha r, the trajectory will be definitely not symmetric.click here to see the code 

Here is a gif simulate the motion of ball in elliptical table.We use vpython to achieve this.

Below is the phase-space plots for different value of alpha.click here to see the code 

We can get the simillar conclusion with the trajectory when alpha =0 and 0.01.In the phase-space plots we can see a gradually varied situation.Even the stadium billiard have a alpha = 0.001 error, the result tend to be regular.We can also do this to the trajectory figure, make alpha =0.001 and 0.1 to see the gradually varied situation.

Now let's talk about our problem.I change alpha to see the Lyapunov exponent.click here to see the code 

It is obviously that with the increase of divergence of alpha, the two trajectory seperate faster.

References

Shan Tan's work

Zongmeng Yang‘s work’