WOLFRAM|DEMONSTRATIONS PROJECT

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.