Monte Carlo analysis of the Lorenz equations – Part II

Cont'd from here.

The non-linearity in the model is clear from this scatter plot. For small perturbations in the inputs from the mean values (depicted as the square dots below), the solutions can end up being anywhere in the space depicted by the scatter plots.

Stated in Lorenz’s words, a flap of the butterfly’s wings for the input conditions can create a drastic change in outputs, possibly resulting in a tornado. In Meteorological terms, this is one of the main reasons why longer term weather forecasts remain a difficulty. Even a statistical analysis such as the one performed here is worthless in predicting the longer term effects of weather, since conditions such as temperatures can lie anywhere in the contour of the scatter plots and thus mean nothing in terms of accurate predictions. This is the crux of Lorenz’s work; that accurate longer term predictions cannot be made when it comes to weather. Further, he leaves the question of the butterfly creating a tornado in a different hemisphere of the world unanswered.

Predictability of phenomena is more reliable for linear systems. While short term predictions can be made for non-linear systems, exact longer term predictions are both unreasonable and impossible. All that one can predict precisely about the location of the electronic butterfly in response to a tap on the jar is that it can be found some place inside the bottle after a few seconds. In a similar (and a rather cynical) vein, all that the Meteorologist can predict with certainty for longer term forecasts is that it will be summer in July in the Northern hemisphere. It only takes a mental extrapolation to extend this analogy to even longer term predictions with climate change, and the reasons for the discussions that ensue!  

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.

 

Edward Lorenz and the Butterfly Theory

Predictability; Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? This was the question proposed by Dr. Edward Lorenz at the 135^th Meeting of the American Association for the Advancement of Science in 1972. Dr. Lorenz can be stated as one of the pioneers of modern day Meteorology, and laid the foundations for equations in weather prediction. He went on to research on whether the behavior of atmosphere is “unstable” with respect to small perturbations in amplitudes when it comes to long term weather predictions.

In his other seminal work, “On the Prevalence ofAperiodicity in Simple Systems”, Lorenz states that weather predictions can be likened to a set of non-linear differential equations, whose solutions are heavily governed by the initial values. He states that periodic solutions would be naïve in weather forecasting, since if there were indeed a periodicity, the statistical prediction would be a trivial one.

It turns out that Lorenz started out with solving a set of 12 differential equations and stumbled upon the aperiodicity in the solutions with minor changes to input conditions. He noticed that re-solving the equations with small differences in the initial values resulted in vastly different outputs over a period of time. To quote Lorenz, “These perturbations were amplified quasi-exponentially, doubling in about four simulated days, so that after two months the solutions were going their separate ways”. Meaning, the instability in the solution was “due to the lack of its periodicity”. Lorenz’s work on non-periodic equations for weather predictions has been the founding blocks for Chaos Theory and the “Butterfly Effect” . 

To be cont'd. 

Implicit ODE’s, Rosenbrock Formulation & the Josephson’s Junction:

ODE’s of the form

are referred to as implicit ODE’s. While a matrix inversion can be used to convert these to an explicit ODE, the form of the Rosenbrock integration scheme can be modified to directly solve for an implicit 1^st order set of ODE’s.

The complete integration can be performed using the same procedures described earlier.

Getting back to numerically solving the Josephson’s Junction equations, which is an implicit ODE, the above formulation can also be used as an alternative and avoid the matrix inversion mentioned earlier.