Piecewise Linear Models for the Quasiperiodic Transition to Chaos
Date
1995-02-24
DOI
Authors
Campbell, David K.
Galeeva, Roza
Tresser, Charles
Uherka, David J.
Version
OA Version
Citation
1995. "Piecewise Linear Models for the Quasiperiodic Transition to
Chaos," chao-dyn/9502018. http://arxiv.org/abs/chao-dyn/9502018
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
quasi-periodic 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.