Finding Maxima and Minima for the Lorenz Attractor

σ

16

r

45.92

b

4

minima

maxima

This Demonstration finds the maxima and minima of the Lorenz system:

dx

dt

=σ(y-x),

dy

dt

=rx-xz-y,

dz

dt

=xy-bz.

For various selections of the model parameters

σ

,

r

, and

b

, you can observe periodic behavior, period doubling, or chaotic behavior. For example,

σ=16

,

r=45.92

, and

b=4

shows chaotic behavior, while

σ=19.8

,

r=56

, and

b=1

gives periodic behavior.

The maxima and minima are easily determined using the built-in Mathematica 9 function WhenEvent. Finally, once the maxima or minima are found, a relatively straightforward extension of the present code allows the determination of all types of bifurcation diagrams.