Tuesday, May 12, 2015

Numerical solution of the Rossler Equations



Much like the Lorenz equations and its associated non-linearities, the Rossler equations also present an “attractor” phenomenon that is used to model equilibrium in chemical reactions. The Rossler ODE’s as presented in Solving ODE’s in R by Soetaert et al are

 
 

The initial conditions are {1,1,1}. Plotted below is the 3D phase plane of the numerical solution. 





 

Tuesday, May 5, 2015

A note on Linear Superposition of functions



Linear algebraic functions can be expressed as a sum of two other functions, given certain constraints. These constraints are typically the boundaries of the original function, from which the two secondary functions can be derived.

Consider a function f(x), defined in the interval [a,b]. The linearity theory helps define two additional functions such that
Several functions can be derived for f1(x) and f2(x) that satisfy the above condition. Considering the simplest linear case, a value for the constant (delta) can be derived with a few generalized assumptions. Each of the functions f1(x) and f2(x) can be constructed such that it meets one of the boundary requirements individually. Namely,


Substituting either one of these conditions for f1(x) or f2(x) into the original equation for f(x), an expression for the constant can be obtained: 



The original function f(x) can then be constructed as a linear sum of f1(x) and f2(x). An example is shown below for one such construction.This concept of linear superpositions come in handy while solving boundary value ODE’s.

 

Tuesday, April 28, 2015

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!  

Monday, April 20, 2015

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.