Lorenz Attractor

Edward Lorenz's famously beautiful butterfly is expressed very simply in matrix form:

   _
  dy      ( -8/3    0   y2 )   _         _
 ----  =  (    0  -10   10 ) * y ;       y = ( y1 y2 y3 )
  dt      (  -y2   28   -1 )

If the matrix is denoted A, and the vector y, you can try integrating with a scheme like Forward Euler;

while ( t < T )
  dydt = A*y;
  y = y+dt*dydt;
  t = t+dt;
end

However you will discover that Euler's method is particularly unsuited for this kind of problem. Very small values of dt are necessary, and even if you manage to get accuracy for a little while, very small differences of initial y quickly lead to hugely different answers. Indeed, this quality makes Lorenz's attractor particularly interesting. It follows a path which is bounded, but not periodic (it never "settles down", or converges). For this reason, the Lorenz attractor is known as a strange attractor.

Matlab comes with a demo of the Lorenz attractor; just type
     lorenz

More Info at:
Chaos Theory