In 1963 Edward Lorenz derived a simple set of equations describing convection in the atmosphere:
(see attached file for equations).
Even though these equations are simple and deterministic, long-term behavior of solutions for some particular values of parameters (e.g. omega = 10, ro = 28, beta = 8/3) could be highly unpredictable. Small variations of initial conditions could result in drastic difference of the corresponding solutions of the system. The latter phenomenon is known as the butterfly effect: small perturbations of the atmosphere caused by the butterfly wings at one location on Earth can result (according to the model) in substantial changes in the atmosphere at another location.
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An example of calculating the difference for different Delta t and comparing the results in plots is shown in ...
In a Matlab script, we demonstrate the application of the Runge-Kutta numerical method for a Lorenz attractor, the Butterfly Effect caused by a small change of initial conditions, and the dependence of the Butterfly Effect on the step of the integration.