The Lorenz system is a system of ordinary differential equations (in 3D). As such, the system is deterministic: given the initial position of a particle, whose motion is governed by the equations of the Lorenz system, the future path is completely determined by the solutions of the equation. However, we discussed in class that predicting the behavior of the Lorenz system is essentially a game of random guess, that is the system seems to behave in a random way, rather than deterministic. In class, we observed this apparently random feature in the motion of the chaotic waterwheel, which is modeled by the Lorenz system. For the waterwheel (see here) it is essentially impossible to predict when the wheel changes direction of rotation, or how many consecutive rotations in the same direction the wheel performs before turning the other way. How do you see this apparently random behavior in the solution curves of the Lorenz system? Use this webpage to reproduce the Lorenz system. Experiment with different values of and include, in your explanation, the plots obtained with 4 different values of .

https://itp.uni-frankfurt.de/~gros/Vorlesungen/SO/simulation_example/