# Euler, Improved Euler, and Runge Kutta for Lorenz attractor, and Butterfly Effect

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In 1963 Edward Lorenz derived a simple set of equations describing convection in the atmosphere:

See attached for equations.

1. Please implement Euler integration scheme for (1) with integration step delta = 0.0001

2. Please implement improved Euler integration scheme for (1) with delta = 0.0001

3. Please implement Runge-Kutta integration scheme for (1)) with delta = 0.0001. Change x(0)=0 to x(0)=0.0001. Observe the butterfly effect.

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##### Solution Summary

In a Matlab script, we demonstrate the application of Euler, Improved Euler, and Runge Kutta numerical methods for Lorenz attractor, and the Butterfly Effect caused by a small change of initial conditions. The solution helps implement the Euler integration scheme and the Runge-Kutta integration scheme for various equations.

##### Solution Preview

The attached script shows an example of how the calculations for 1, 2, and 3 may be arranged.

IMPORTANT TO NOTE:

(1) To speed up the tests, I took delta_t and delta_x larger than that suggested in the text.

When you arrive to the final ...

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