ABSTRACT
We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode locking and the quasiperiodic transition to chaos. For instance, for these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. Our main results describe (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation interval; and (3) the structure and bifurcations of the attractors in one of these families. We discuss the interpretation of these maps as low-order spline approximations to the classic "sine-circle" map and examine more generally the implications of our results for the case of smooth circle maps. We also mention a possible connection to recent experiments on models of a driven Josephson junction. (c) 1996 American Institute of Physics.
ABSTRACT
A model of magnetic field generation by the turbulent motions of a highly conducting fluid is constructed. It is assumed that the field is generated in individual cells of space in a statistically independent manner. A coupling between the cells is realized by magnetic diffusion. The rates of growth of the field and its moments are calculated. The field distribution generated by this mechanism possesses an intermittent nature in both time and space.
