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Programs
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Lorenz Equation
Launch Lorenz Attractor Launch Small Screen Version
Even though the Lorenz equations appear to be fairly simple differential equations, the solution of these equations by iteration can demonstrate many of the principals of chaotic systems. The differential equations are:
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In these equations x, y, z are state variables. Their initial values (initial conditions) must be specified by the user. Sigma, r and b are not variables but rather parameters. They may be manipulated before a time evolution is run. This manipulation can change the shape of the resulting trajectories. |
The Lorenz equations are used to model the weather. Their most well known chaotic attractor is often called the Lorenz Butterfly, in association with the Butterfly Effect -- another name for the Sensitivity to Initial Conditions (SIC) which is the defining characteristic of chaotic systems.
To see the Butterfly Effect in action start off the Lorenz equations with Multiple Trajectories enabled. Make the deviation (which represents SIC) small and watch how after a while of following each other closely the separate trajectories break off and act totally independent.
Some other simple activities are:
Notice how the default values for a single trajectory generate the "butterfly."
If you lower r, to say 22, it will graph a fixed point for the same a and b settings.
If you raise r to 330 or some high value you will begin to see the Lorenz limit cycle.