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.
(see attached file for full description)
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.