Aperiodic flow-induced oscillations of collapsible tubes:
a critical reappraisalC.D. Bertram a,
, J. Timmer b, T.G. Mu¨ ller b, T. Maiwald b, M. Winterhalder b,
H.U. Voss b
a Graduate School of Biomedical Engineering, University of New South Wales, Sydney, NSW 2052, Australia
b Freiburger Zentrum fu¨ r Datenanalyse und Modellbildung, Fakulta¨t fu¨ r Physik, Albert–Ludwigs–Universita¨ t Freiburg, Eckerstraße 1, 79 104
Freiburg, Germany
Received 7 April 2003; received in revised form 14 October 2003; accepted 21 November 2003
Abstract
The evidence for the aperiodic self-excited oscillations of flow-conveying collapsible tubes being mathematically chaotic is reexamined.
Many cases which powerfully suggest nonlinear deterministic behaviour have not been recorded over time-spans which
allow their exhaustive examination. The present investigation centred on a previously recorded robust and generic oscillation, but
more recent and more discerning tests were applied. Despite hints that a low embedding dimension might suffice, the data
appeared on most indices high-dimensional. A U-shaped return map was found and modelled using both radial basis functions
and polynomials, but lack of detailed structure in the map prevented effective parameter estimation. On the basis of power-law
rather than exponential divergence of nearby trajectories, and of inability to discriminate against behaviour which would also be
manifested by a surrogate consisting of a noise-perturbed nonlinear periodic oscillator, it is concluded that the data do not support
the idea that the aperiodicity in the particular oscillation examined is caused by deterministic chaos. There was evidence that
the distributed nature of the physical system might underlie aspects of the high dimensionality. We advocate equally searching
testing of any future candidate chaotic oscillations in the investigation of collapsed-tube flows.
# 2003 IPEM. Published by Elsevier Ltd. All rights reserved.
Keywords: Nonlinear dynamics; Deterministic chaos; Time-series analysis; Self-excited oscillation
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