This page leads you through the main features of the applet. You can also review the background and mathematical details.

The parameters of the Lorenz Model are a,b and c. The most important parameter is a, and usually b and c are set to the values b=8/3 and c=10. In the context of convection in a fluid or the atmosphere, a is proportional to the temperature difference across the layer responsible for driving the fluid motion at a rate given by the variable X. (Y and Z then tell us about the changes in the temperature distribution in the layer due to the heat carried by the moving fluid.)

The "standard" value of a is a=28. Start the applet at these default values by hitting Reset and then Start. You will see a plot of Z(t) against X(t) as t varies. The familiar Lorenz butterfly appears. Notice both the complexity of the plot (the "randomness" or "unpredictability") but also the definite structure - this combination is characteristic of chaos.

A number of parameters can be changed to give the best animated plot, depending on the speed of the computer etc. The variable dt gives the time stepping in the numerical integration. Small values may make the animation smoother but slower. Beware: too large a value will give inaccurate answers - there is currently no accuracy check in the numerical scheme! The variable tran sets a number of time steps over which the equations are evolved first to eliminate transients and then to construct a "phantom image", which also sets the scale of the plot to eliminate the continual rescaling at early times. You can set tr an=0 to eliminate this effect. The speed scrollbar sets the speed of the animation (roughly, updates per second), and can be tuned to smooth the animation.

The full dynamics needs to show the three variables X,Y,Z as they vary in time. On the computer screen it is easiest to show various projections such as the Z-X projection just seen. The two variables plotted are chosen by the x-ax and y-ax parameters with 1 standing for X, 2 for Y and 3 for Z. Try out various combinations to look at different aspects of the full 3-d plot. (Remember to hit Enter or the Reset button after changing a value.)

Choosing x-ax=0 plots the time variable along the x-axis. This can be used to look directly at X(t) etc.

The starting values of the three variables are set by X0, Y0, Z0. Try out different values. If you have tr an=0 you will see different behavior at early times (the transient). However the dynamics evolves towards the same butterfly structure, which is therefore called an attractor. The dynamics on the attractor will be similar in outline, but different in details depending on the initial conditions, i.e. chaotic. The attractor is therefore called a strange attractor.

Perhaps the defining property of chaos is the sensitive dependence on initial conditions or poetically the butterfly effect. This is demonstrated by plotting two solutions together starting from very slightly different initial values. The small difference between the initial values is set by dX0, dY0, dZ0.

It is also interesting to study the dependence on parameters particularly the behavior as a function of a. (You might want to set tr an=0 for this investigation.)

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Last modified 18 August, 2009
Michael Cross