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Discrete Dynamics in Nature and Society
Volume 5 (2000), Issue 2, Pages 107-120

Staircase baker's map generates flaring-type time series

Division of Theoretical Chemistry, Auf der Morgenstelle 8, University of Tübingen, Tübingen 72076, Germany

Received 27 December 1999

Copyright © 2000 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The baker’s map, invented by Eberhard Hopf in 1937, is an intuitively accesible, two-dimensional chaos-generating discrete dynamical system. This map, which describes the transformation of an idealized two-dimensional dough by stretching, cutting and piling, is non-dissipative. Nevertheless the “x” variable is identical with the dissipative, one-dimensional Bernoulli-shift-generating map. The generalization proposed here takes up ideas of Yaacov Sinai in a modified form. It has a staircase-like shape, with every next step half as high as the preceding one. Each pair of neighboring elements exchanges an equal volume (area) during every iteration step in a scaled manner. Since the density of iterated points is constant, the thin tail (to the right, say) is visited only exponentially rarely. This observation already explains the map's main qualitative behavior: The “x” variable shows “flares”. The time series of this variable is closely analogous to that of a flaring-type dissipative dynamical system – like those recently described in an abstract economic model. An initial point starting its journey in the tale (or “antenna”, if we tilt the map upwards by 90 degrees) is predictably attracted by the broad left hand (bottom) part, in order to only very rarely venture out again to the tip. Yet whenever it does so, it thereby creates, with the top of a flare, a new “far-from-equilibrium” initial condition, in this reversible system. The system therefore qualifies as a discrete analogue to a far-from-equilibrium multiparticle Hamiltonian system. The height of the flare hereby corresponds to the momentary height of the H function of a gas. An observable which is even more closely related to the momentary negative entropy was recently described. Dependent on the numerical accuracy chosen, “Poincaré cycles” of two different types (periodic and nonperiodic) can be observed for the first time.