Monte Carlo analysis of the Lorenz equations – Part I

Cont'd from here.

Solving a differential equation on a computer in the early 70’s was by no means an easy task, let alone solving a set of 12 aperiodic coupled non-linear ODE’s. Lorenz states that he got a break when he met with Dr. Barry Saltzman, whose work on thermal convection resulted in a set of 7-equations, 4 of which approached zero. Lorenz was able to modify the other three equations and solve for the aperiodicity of the outputs. These three equations are commonly referred to as the Lorenz model.

How does one study the aperiodicity of the Lorenz model? By following Lorenz’s lead and introducing minor perturbations to the initial values. And that is exactly where the Monte Carlo technique comes in useful. As stated earlier, the MC technique works on the premise of allocating randomized probabilities for perturbed inputs and studying the behavior of the system. The inputs used for the process were as follows:

Randomized initial values were thrown as inputs to the model within the bounds shown above. After 2000 such throws, the outputs were obtained for a specific instant of time, 15 time units in this case. The results of the scatter phase-plots are shown below.To be cont'd.