| The first part of this thesis deals with the analysis of a small array of Josephson junctions, superconducting devices of markedly nonlinear behavior. The components of the array are modelled as resistively-shunted junctions and are driven by direct currents. A chapter is provided which reviews the validity and features of such a model for the case of a single junction. This chapter also includes background information on the subjects of bifurcation, chaos, and fractals.; In the following chapters, the array of three junctions is studied, first with one driving current and later with an additional bias current. The analysis includes both numerical results from computer simulations and analytic computations using a perturbative approach. The two approaches are shown to be in good agreement.; The behavior of the array is dominated by hysteresis effects. With a single driving current, four branches are seen in the current-voltage characteristics. The functional dependences of these curves, as well as the hysteretic cutoffs, are calculated. Small regions of bifurcations and chaotic behavior are also seen and examined using the ideas of the first chapter.; The addition of a bias current leads to a countable number of periodic trajectories as well as quasiperiodic motion. Both the quantitative form of these orbits and the basins of attraction for them are predicted using the perturbative technique. The extension of this technique to related models is discussed, and comparison of the results with available experiments is provided.; The second part of the thesis describes an exact bound-to-site transformation for diffusion-limited aggregation. A review is provided which summarizes the field of aggregation and which demonstrates the need for such exact results. The equivalence maps a class of partial adhesion problems on arbitrary lattices to absolute adhesion problems on transformed lattices. Examples are given for diffusion in the presence and absence of an external field. The correspondence relates two conjectured aspects of universality in the scaling behavior, namely the irrelevance of lattice structure and adhesion probability. |
