Lorenz Model

Abstract

   The two model that we have considered so far in the chapter--the pendulum and  the logistic map--are both very simple systems, yet they exhibit extremely rich behavior. It is not surprising that other slightly complicated systems are also capable of chaotic behavior. When we chaotic and unpredictable behavior,an example that naturally comes to mind is weather. Because of the economic importance of having accurate weather predictions, a good deal of effort has been devoted to the problem.While much of this effort has also been devoted to understanding the weather problem from a more fundamental point of view. It was work of this atmospheric scientist E.N.Lorentz that provided a major contribution to the modern filed of chaos.

Background

   Lorentz was studying the basic equations of fluid mechanics, which are known as the Navier-Stokes equations and which can be regarded as Newton’s law written in a form appropriate to a fluid. The specific situation he considered was the Rayleigh-Benard problem, which concerns a fluid in a container whose top and bottom surfaces are held at different temperatures. Indeed, he grossly simplified the problem as he reached to the so-called Lorentz equations, or equivalently, the Lorentz model, which provided a major contribution to the modern field of physics.

Main Body

The lorenz model

The Lorentz variables x,y,z a,r,b are derived from the temperature, density and velocity variables in the original Navier-Stokes equations, and the parameters are measures of the temperature difference across the fluid and other fluid parameters.

Euler Method

Euler-Cromer method is designed for second-order differential equations, so it is not directly applicable to the Lorentz model.. Here, the Euler algorithm can actually be used to treat the Lorentz problem.

Here is the code

   This is the result of variation of the Lorentz variables y,z as a function of time. The calculation is performed using the Euler method with a time step dt=0.001and other parameters set as a=10,b=8/3. The initial conditions are .X0=1;Y0=0;Z0=0

   As for r=5,10,15, the behaviors are similar: there is an initial transient, and after it decays away,y,z are constant independent of time. The transient takes longer as r increases. These cases correspond to steady convective in the original fluid, analogous to the regular chaotic motion of the pendulum. As for r=25, the initial transient is roughly periodic, but it gives way to irregular. This corresponds to the chaotic condition.

Chaotic Behavior r=25

we have three variables in Lorentz model, projection of the trajectory can be considered.

To show it more directly, a 3D image is made to show the trajectory.

                           The Billiard Problem

Abstract

   We can think of this as a billiard ball that moves without friction on a billiard table. The ball is given initial velocity, and the problem is to calculate and understand the resulting trajectory. This is known as the stadium billiard problem.

The code for square table

Of course we can consider another kind of table, like circle table

The code for circle table

Reference

Thanks for Tanshan's code. He did sinish his work very well every time.

And always the acknowledgement fo textbook