Piecewise Linear Models for the Quasiperiodic Transition to Chaos

Copyright © (1996) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics

*** NOTE - For full text of paper contact Charles Tresser (tresser@watson.ibm.com).*** 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) teh 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.

By: David K. Campbell (Univ. Ill., Urbana), Roza Galeeva (ENS Lyon, Fr.), Charles Tresser and David J. Uherka (Univ. of N.D., Grand Forks)

Published in: Chaos, volume 6, (no 2), pages 121-54 in 1996

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