Collective induced phenomena in systems of coupled oscillators

• Main Author: Mulhern, Colm
• Other Authors: Burbanks, Andrew David ; Hennig, Dirk ; Osbaldestin, Andrew Harold
• Published: University of Portsmouth 2012
• Subjects: 621.381533; Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.570676

This thesis will explore the deterministic dynamics of systems of coupled oscillators. In particular, the focus of the thesis will be concerned with the transport properties of these systems. Of interest is how particles work together, cooperatively, to achieve directed transport. For this reason, the strength of the coupling between the particles will serve as the main control parameter. Further, ensemble dynamics will serve to highlight some of the collective effects of these systems. The thesis will be split into two parts. The first part will look at a class of autonomous Hamiltonian systems, while the second part will consider a class of driven and damped systems. A common feature of these systems is that they contain a spatially open component that will facilitate long range transport. More precisely, transport proceeds in a spatially symmetric and periodic multiple well potential. Thus transport will be characterised by particles overcoming successive energetic barriers created by the potential landscape. The cooperative effects between the particles will become apparent in Part I when autonomous Hamiltonian systems are considered. In the uncoupled limit the full systems decompose into two integrable subsystems and the dynamics are fully understood. However, the dynamics become more complicated when the particles are coupled. As these systems are conservative a coordinated energy exchange between the particles is often required for directed transport to ensue. Interestingly, these systems contrast well with the nonautonomous one and a half degree-of-freedom Hamiltonian systems, where transport occurs through intermittent periods of directed motion in so-called ballistic channels. The autonomous two degree-of-freedom counterpart considered here relies on a rather different mechanism for directed transport that will be provided solely by regular structures in phase-space. With the inclusion of external driving and damping (Part II) the transport dynamics are controlled by various coexisting attractors in phase-space. The nature and stability of these attractors is determined by the system parameters. As before, cooperative e�ects will play a key role when it comes to particle transport. Notably, it will be seen that coupling between the particles can result in a suppression of chaos that will allow for, for example, collective periodic motion of rotational type. Particular attention will be paid to the phase-space structures and how they change as the coupling parameter is varied. Throughout the thesis these nonlinear systems, and their transport features, will be explored using analytical and numerical means. A number of model systems will also be introduced to further illuminate the systems dynamics.