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VOL. 37 | 1999 Stability and shadowing in convex discrete-time systems
Phil Diamond, Alexei Pokrovskii

Editor(s) Tim Cranny, Bevan Thompson

Abstract

In the realization of a dynamical system on a computer, all computational processes are of a discretization, where continuum state space is replaced by the finite set of machine arithmetic. When chaos is present, the discretized system often manifests collapsing effects to a fixed point or to short cycles. These phenomena exhibit a statistical structure which can be modelled by random mappings with an absorbing centre. This model gives results which are very much in line with computational experiments and there appears to be a type of universality summarized by an Arcsine Law. The effects are discussed with special reference to the family of mappings $f_e(x) = 1 - |1 - 2x|^l, x \in [0, 1], 1 \lt l \leq 2$. Computer experiments display close agreement with the predictions of the model.

Information

Published: 1 January 1999
First available in Project Euclid: 18 November 2014

zbMATH: 1193.37031

Rights: Copyright © 1999, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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