Study of the Dynamic Behavior of the Lorenz System

σ

16

r

45.92

b

4

option

time-series

phase-space

power spectrum

autocorrelation function

This Demonstration presents the dynamic behavior of the Lorenz system:

dx

dt

=σ(y-x),

dy

dt

=rx-xz-y,

dz

dt

=xy-bz.

For a particular selection of model parameters

σ

,

r

, and

b

, you can observe periodic behavior, period doubling, or chaotic behavior. The Demonstration illustrates several important concepts of nonlinear dynamics, such as the time-series plot, the phase-space diagram, the power spectrum, and the autocorrelation function plot. In addition, an estimate of the maximum Lyapunov exponent is displayed for selected model parameters.

For

σ=16

,

r=45.92

, and

b=4

, you can observe chaotic behavior, which is confirmed by the power spectrum diagram. The phase-space diagram is that of a strange attractor. In addition, the estimate of the maximum Lyapunov exponent is close to 1.49. A positive Lyapunov exponent is further indication of chaotic behavior.

For

σ=19.8

,

r=56

, and

b=1

, you can observe periodic behavior, which is confirmed by the power spectrum diagram. The phase-space diagram is that of a limit cycle. In addition, the estimate of the maximum Lyapunov exponent is approximately equal to zero. Thus, all Lyapunov exponents are less than zero, which is a further indication of periodic behavior.