The Lorenz Model - Demo 5

Parameter Dependence

We will often be interested in how chaos develops from simpler dynamics as a parameter of the system is changed. The behavior of the Lorenz model is quite complicated - in other examples we will find that quantitative predictions can be made. Here we will just look at qualitative trends. The parameter a corresponds to the control parameter r of the Lorenz model.

First let us remind ourselves of the familiar behavior:

There are actually time independent solutions corresponding to steady fluid flow, but these are unstable for the parameters we are using. Here is a plot starting from an initial condition close to a time independent solution (known as a fixed point):

The orbit spirals away from the initial point, and eventually ends up on the familiar butterfly.

If we change the parameter a to 20 however:

the orbit spirals in to the fixed points which are now stable - the long time state would be stationary.

Depending on the initial values a "chaotic transient" might be seen, for example at a=22 and for initial values X=5, Y=5, Z=35 the orbit begins to spiral into the fixed point for time t>40 or so:

For a>24.06 the chaotic motion and the stationary fixed points are both stable solutions - which one represents the asymptotic behavior depends on the initial conditions. Since the chaotic state persists at long times, and different initial conditions lead to this state, this portion of phase space is called a "chaotic attractor" or a "strange attractor". We will be studying the geometry of these strange attractors later. Here are initial conditions leading to the chaotic state:

By changing the values of X0,Y0, or Z0, or by starting from initial values set by a mouse click on the running plot, you should be able to find orbits that spiral into one or other of the fixed points. Incidentally the fixed points correspond to a pair of counter-rotating convection rolls in the convection system that motivated the Lorenz model: the two fixed points correspond to the two possible rotation directions. The chaotic motion is then the random reversal of the sense of rotation.

For a>24.74 the chaotic attractor collides with the fixed points, which then become unstable. This then gives the behavior Lorenz studied.

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Last modified Thursday, November 18, 1999
Michael Cross