Exercise 10: The Lorenz Attractor

Here are your 5 bonus points   Credits: 5/5

To extract the exercise files for this weeks exercise, do

    cd ~/ComputerPhysics
    cvs update -d

In case of problems, see the CVS update help page.

To avoid problems with "undefined procedures or functions" make sure IDL can find the necessary procedures.

• Get an example of a solution to the Lorenz problem by running the lorenz.pro procedure.
  • NOTE: You can copy/paste from this window, but it may be a better exercise to type the commands manually.

    IDL> f=lorenz()
    IDL> help,f                ; check what type of variable the answer is
  

• Plot the time behavior of single coordinates

    IDL> plot,f[0,*]
    IDL> plot,f[1,*]
  

• Plot the xy and xz projections

    IDL> plot,f[0,*],f[1,*]
    IDL> plot,f[1,*],f[2,*]
  

• Set up two 3D perspective transformations and show / animate the result

    IDL> tr=setup(az=30,ax=20,persp=3)
    IDL> showtail,f,tr
    IDL> showtail,f,tr,wait=0.02  ; slower (0.005 sec default)
  

• Note that, by chance, the solution almost retraces earlier orbits, towards the end. The default solution has 1000 steps, so if you make one with 1500 steps you have a chance to see the solutions diverge again:

    IDL> f=lorenz(1500)
    IDL> tr=setup(az=30,ax=20,persp=3)
    IDL> showtail,f,tr,wait=0.01 
  

• Calculate two solutions, from two slightly different initial values, and compare one of the coordinates.

    IDL> f1=lorenz(f0=f0)
    IDL> f2=lorenz(f0=1.0001*f0)
    IDL> plot,f1[0,*]-f2[0,*]
    IDL> plot,f1[0,*]
    IDL> oplot,f2[0,*],line=2
  

• The Lyapunov exponent lambda characterizes the average exponential growth. Try a few, with

    IDL> f1=lorenz(t=t)                               ; t is returned
    IDL> plot,/ylog,t,abs(f1[0,*]-f2[0,*])
    IDL> lambda=...                                   ; take your pick
    IDL> oplot,t,1e-4*exp(lambda*t),line=3            ; try it out
  

  How large is lambda, roughly? 0.7 1.0 1.4 Credits: 3/-1 OK

IDL Object Graphics provides high level graphics functionality, such as interactive 3-D visualization, with a very small programming effort. Detailed documentation is available though the IDL online help system, but you do not need it to carry out this exercise.

Example 1:

Open the file viz3d_simplest.pro from IDLDE, compile it and run it. With just three lines of code it produces an interactive display of the Lorenz attractor.

• The expression f=lorenz() produces the {x,y,z} coordinates, in the form of an array dimensioned [1000,3].
• The call to obj_new('IDLgrPolyLine',data=f) returns a "polyline" graphics object representing the coordinates in 3D.
• The call to xobjview produces an interactive display showing the object.
  • Resize the window to a convenient size.
  • Use the three toolbuttons rotate, zoom, and pan (identified by tooltips when you point at them), and the mouse, to rotate, zoom, and pan!

    IDL> xobjview,obj_new('IDLgrPolyLine',data=lorenz()) 

Example 2:

Open the file viz3d_simple.pro from IDLDE, compile it and run it. Consider how it differs from the previous one.

• The call to obj_new('IDLgrModel') produces an abstract 'container' to which graphics objects may be added.
• The call to the object method m->Add, o adds the polyline from the previous example to it.
• The call to the object method m->Add, bbox(f) adds a bounding box, constructed by the bbox.pro procedure.

Example 3:

Open the file viz3d.pro from IDLDE, compile it and run it. Consider how it again differs from the previous one.

• The color of the background is changed.
• The calls to o->SetProperty, color=[R,G,B] set the color of the objects (using red, green, and blue values between 0 and 255).