The lorenz model: A simple chaotic problem

郭帅斐

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The summery:

Lorenz model is another model that can go to chaos, which is put forward to explain the change of the weather. As the last program, we just run the program, and we find that the system goes to chaos. This time, the phrase diagrams and the attractors are more complicated but beautiful.

the psudocode:

We get the solution of the diffential equation theough numerical way. The lorenz quation are as follows:

$$\frac{dx}{dt}=\sigma (y-x)$$
$$\frac{dy}{dt}=-xz+rx-y$$
$$\frac{dz}{dt}=xy-bz$$
We solve this equation numerically, which is very easy, and show the result through a $x-t$ figure and a phase diagram, as well as an attractor figure. Here I use a more powerful way to solve ODE, that is a class defination with Euler method inside.

The result:

• First, I change the value of r, and investigate the amplitude. The result is as follows:
  
  when the parameter $r$ go to 1.2, the syatem go to chaos.
• Second, the phrase diagram of $z-x$ have been plotted as follows:
  
  And a more longer time one:
  
  The figure is just like a butterfly!
• As I have mentioned before, we can explore the attractors of the phrase diagram. Here I plot the attractors in the $x=0$ plane.
  
  It's amazing that such a butterfly-like phrase diagram have a Poincare section like a simple 'Y'.

The conclusion:

In the program, The chaotic phenomenon of lorenz model have been analysed. We get a lot of information from the $x-t$ figure, phrase diagram and the attractors figure.