Averaged Behaviour of Nonconservative Coupled Oscillators

Abstract

In this Thesis we study the dynamics of systems of two and three coupled oscillators by efficiently applying Normal Form theory. The subject of Coupled oscillators plays an important part in dynamical systems. It has a wide range of applications in various fields like physics, biology, economics and many more disciplines. In physics for example, coupled oscillators pop up in mechanical systems, like the ones in this thesis, electronics and lasers. In biology, most organisms seem to be coupled through periodicities in our environment related to the motions of the sun, earth and the moon. Studying and classifying the behaviour of coupled oscillators is therefore of great importance. In Chapter 1 we start the study of a two degrees of freedom mechanical system excited parametrically. First the system is averaged, then an elaborate stability analysis of both the trivial equilibrium and the periodic solutions is presented. A comparison is given between the averaging method and the asymptotic method called "Harmonic Balance ". Both methods yield results that are in good agreement. In Chapter 2 the same system is studied, only now a global bifurcation study is done. This system turns out to have many "large" attractors, some of them reminiscent of the Lorenz attractor, interacting with each other. Phenomena like torus destruction and the emergence of strange attractors are studied and analysed numerically. In Chapter 3 we study a three degrees of freedom self-excited mechanical system using averaging and give an answer to whether the single mode always prevails. In Chapter 4 a self-excited system in 1:2:3 internal resonance is studied. Double Neimark-Sacker bifurcations were easily detected and a stable T3 torus was numerically traced back to the original system. We discovered the averaged system has a period doubling curve exhibiting behaviour never reported before in the literature. This nongeneric phenomenon should be subject of further study