Get a grip on chaos : Tailored measures for complex systems on surfaces
Abstract: Complex systems are ubiquitous in physics, biology and mathematics. This thesis is concerned with describing and understanding complex systems. Some new concepts about how large systems can be viewed in a lower dimensional framework are proposed. The systems presented are examples from ecology and chemistry. In both cases we have a large amount of interacting units that can be understood by focusing on more abstract featuresthat are the result of internal interactions.The predator-prey system investigated consists of ground beetles, Pterostichus cupreus L. (Coleoptera: Carabidae), that feeds on bird-cherry oat aphids. The beetles' movement can consistently be described by a combined model of surface diffusion and biased random walk. This allows conclusions about how fast and in which fashion the beetle covers its habitat.Movement is dependent on aphid densities and predation, in turn modifies aphid distributions locally. The presented generalized functional response theory describes predation rates in the presence of spatial heterogeneity. A single measure for fragmentation captures all essential features of the prey aggregation and allows the estimation of outbreak densities and distributions.The chemical example is the catalytic oxidation of CO on a Pt(110) single crystal surface. Unstable periodic orbits reconstructed from experimental data are used to reveal the topology of the attractor, underlying the time series dynamics. The found braid supports an orbit which implies that the time series is chaotic.The system is simulated numerically by a set of partial differential equations for surface coverage in one space dimension. The bifurcation diagram of the corresponding traveling wave ODE reveals the homoclinic and heteroclinic orbits that organize the phase space and mediate the transition to chaos. Studies in the PDE-framework relate this to the stability and to the interaction of pulse-like solutions.
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