Physique | Technique
Toby Lane, 2004 | Gryon, VD
A new method to detect the presence of chaotic trajectories in a deterministic system is presented. The work describes how this method is successfully applied to the driven pendulum in three steps: (I) analysing the characteristics of chaotic solutions; (II) studying the parameter sets that lead to these solutions; and (III) developing a numerical method to identify them. The equation of motion of the driven pendulum is solved numerically and the chaotic solutions are classified into four different types. Results show that the system is strongly dependent on energy, and an energy balance is essential for a solution to be chaotic. Chaotic parameter sets align themselves along cluster planes within the three-dimensional parameter space. The developed detection method is found to occasionally miss chaotic solutions. However, comparisons with other methods in the literature suggest that the developed method is comparable in terms of accuracy and more efficient in terms of computational time.
The work addresses the application of chaos theory to the driven pendulum. Three questions are asked: (I) how does chaos manifest itself in the system? ; (II) are there clusters that emerge in the parameter sets and lead to chaotic motion? ; (III) with the aim of finding chaotic solutions and parameter sets, how can a numerical method be developed to classify a given solution?
Solutions are found for different parameter sets by solving numerically the differential equation of the driven pendulum. A novel method is developed in MATLAB (TM) to classify the solutions as leading to chaos or not. By wrapping oscillation angles within the range of pi and by differentiating the corresponding solution, a binary series is created that determines whether the solution is periodic or chaotic. The method is applied to 20’000 solutions calculated from parameter sets randomly selected through Monte-Carlo sampling. For 100 randomly sampled solutions, the novel method is compared to classical methods for identifying chaos, Poincaré sectioning and Lyapunov exponents.
The developed method accurately classifies solutions 98% of the time. Poincaré sectioning is just as accurate, and Lyapunov exponents, while more accurate, is two degrees of magnitude slower. Four different types of chaotic solutions are found: Flippers, Steppers, Temperamentals and Chaotic Sinusoidals. Optimal values of chaotic parameters are correlated with the gravity the pendulum is subjected to. In addition, the chaotic parameter sets concentrate around areas named “cluster planes”.
The developed method turns out to be the most efficient in computational cost, while the method based on Lyapunov exponents is the most reliable. Chaotic Sinusoidals and Flippers are the types of solutions that can be difficult to detect numerically because they are less chaotic. Results suggest that a balance between dissipated energy and forcing is crucial to whether the system is chaotic or not. If parameters lead to too strong an external force or either too much or too little dissipation, the system does not display chaotic behaviour. The energy balance is what the cluster planes seem to correspond to: optimal combinations of parameters. The phase space plots obtained for the driven pendulum can be compared to those found in the Lorenz system. The driven pendulum appears to have a sort of ‘strange attractor’ in the phase space plots, with two global areas of attraction similar to the two lobes on the Lorenz attractor.
This research has shown that chaos manifests itself in a driven pendulum in terms of four types of solutions, and that the chaotic parameter sets are organised along cluster planes. A novel method to detect a chaotic solution has been developed. Although not 100% reliable, the method has been shown, for the purposes of this report, to be the most efficient compared to other methods in the literature. Different methods can be combined to increase accuracy. Future work should assess whether the approach to detecting chaos could be more widely applicable, for example to other systems. It should also be emphasised that chaotic systems, while challenging to predict, always remain deterministic. As with the weather, they are much harder to predict at anything other than relatively short time scales.
Appréciation de l’experte
Dr. Maura Brunetti
Dans cette étude, Toby décrit de manière très détaillée et complète les outils numériques et statistiques nécessaires pour établir la nature chaotique des trajectoires d’un pendule forcé. Toby a travaillé de manière autonome et a développé une technique originale pour sélectionner les orbites chaotiques en partant de son intuition, dans un premier temps sans regarder les méthodes déjà existantes dans la littérature scientifique. Il a montré à la fois de la créativité et de la rigueur, deux caractéristiques importantes pour un chercheur.
Prix spécial «MILSET Expo-Sciences International (ESI)» décerné par la Fondation Metrohm
Lycée-Collège de l’Abbaye, St-Maurice
Enseignant: Daniel Erspamer