Abstract

Chaotic systems can be found everywhere in everyday life: Population growth, the weather or even a dripping tap can show a strong sensitivity to initial conditions. However, not all initial parameters of a chaotic system lead to chaotic behaviour to the same extent. The Predictability Time Horizon (PTH) indicates how long it takes for two chaotic systems with similar initial conditions to reach a certain degree of dirvergence. From then on, the system is considered unpredictable in practice for these initial conditions, since a prediction based on the mathematical model would be inaccurate.

A method was developed to estimate the PTH of the double pendulum as a function of the initial conditions. Using this method, any initial angles could be assigned a colour representing the corresponding PTH. By colouring the parameter space, a Mandelbrot-like set could be discovered.

An attempt was then made to predict the PTH for arbitrary initial angles using an artificial feedforward neural network, consisting of eight layers of neurons. On average, the model’s prediction had an error of 1.79 seconds, and when asked to categorise given initial angles as chaotic or non-chaotic, it did so with an accuracy of 96.6%. Finally, a visual comparison between prediction and reality showed that a neural network can be a reliable predictive model for chaotic systems..