Global bifurcations and chaotic dynamics in physical applications

Detta är en avhandling från Uppsala : Acta Universitatis Upsaliensis

Sammanfattning: The aim of this work has been to analyse global bifurcations arising in a laser with injectedsignal and in a catalytic reaction on a surface of Pt, from the point of view of dynamicalsystems theory.The 3-dimensional ordinary differential equation which models the laser was found to contain a homoclinic orbit to a saddle-focus equilibrium, what corresponds to the Šil´nikov phenomenon. It is well known that under certain eigenvalue relationship, this globalbifurcation displays chaotic dynamics. In this problem the fixed point was also involved in a Hopf/saddle-node local bifurcation. The interaction of the Šil´nikov phenomenon and the saddle-node bifurcation was studied by constructing a geometrical model. It was determined that as the homoclinic orbit approaches the saddle-node bifurcation, thechaotic dynamics vanishes. Also "bubbles" of periodic orbits, in the period vs. parameterbifurcation diagram, were analysed in this context.The catalysis model, on the other hand, is an excitable reaction-diffusion equation.This equation, in one spatial dimension, displays a transition to spatiotemporal chaos,where an incoherent collection of pulse-like solutions are found. Homoclinic and heteroclinic orbits in the 3-dimensional traveling-wave ODE were analysed, corresponding topulses and fronts in the PDE traveling with constant velocity. The stability analysis ofthese coherent structures revealed that the pulses, characteristic of excitable media, undergo a bifurcation. It is associated to the break-up of a heteroclinic cycle between tworeal saddle fixed points the ODE. The geometrical model of the interaction of this cyclecoupled to a saddle-node bifurcation of the fixed points, reveals that heteroclinic cycleswith arbitrary number of loops, exist. Also chaotic dynamics, due to a Smale's horseshoe,was found. It is observed that as the saddle-node bifurcation creates the fixed points, allmulti-loop homoclinic and heteroclinic orbits are created in an explosion. This constitutesa novel mechanism to produce chaos.

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