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Elliptic behaviour in the sawtooth standard map

Ashwin, Peter (1997) Elliptic behaviour in the sawtooth standard map Physics Letters A, 232 (6). pp. 409-416.


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This paper examines the standard map with sawtooth nonlinearity when the eigenvalues of the Jacobian lie on the unit circle. This is an area-preserving map of the torus to itself that is linear except on a line on which it is discontinuous. We discuss the closure of the set of images of the discontinuity and present numerical evidence that its Lebesgue measure is positive. Moreover, we present evidence that the measure of the closure of images of the discontinuity changes continuously with the parameter k. This means that the sawtooth standard map may exhibit coexistence of two positive measure subsets on which the dynamics is respectively regular and irregular in a certain sense. In the appendix we show that this map is equivalent to a map studied by electronics engineers as a model for the quiescent behaviour of a linear lossless digital filter with "two's complement" overflow.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Ashwin, Peter
Date : 4 June 1997
DOI : 10.1016/S0375-9601(97)00455-6
Additional Information : This is a pre-copy-editing, author-prepared, peer-reviewed PDF of an article published in Physics Letters A, 232, 409-416. Click here to access the publisher's version. © 1997 Published by Elsevier Science B.V.
Depositing User : Mr Adam Field
Date Deposited : 27 May 2010 14:40
Last Modified : 31 Oct 2017 14:01

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