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Proceedings Paper

Search for chaotic output from a sigma-delta modulator: an archetypal retarded nonlinear system
Author(s): David Thomas Hughes; S. McLaughlin; Gary Ushaw; Bernard Mulgrew; David S. Broomhead
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Paper Abstract

One of the most common methods by which a non-linear physical system can be determined to be stable, globally unstable, or in fact chaotic is by an investigation of its Lyapunov exponents. When the system contains a delay equation the correct co-ordinate space for the system is infinite dimensional. In this case special techniques must be used to make the problem solvable by numerical methods. We have investigated a practical example of such a retarded non-linear system: the continuous Sigma-Delta modulator (CSDM). The CSDM is an adaptation of the widely used digital Sigma-Delta modulator in which, now, the input and output are continuous functions of time, the quantizer has been replaced with a sharp hyperbolic tangent function and the feedback delay time can be varied continuously. This allows the system dynamics to be modeled by delay ordinary-differential equations. We have simulated such a device for various feedback configurations and delay times and have been able to show how a full calculation of the Lyapunov exponent spectrum allows a detailed analysis of stability conditions for the CSDM. Furthermore, we discuss the implications of this approach for retarded non-linear systems in general and a range of signal processing applications in particular.

Paper Details

Date Published: 18 November 1993
PDF: 11 pages
Proc. SPIE 2038, Chaos in Communications, (18 November 1993); doi: 10.1117/12.162673
Show Author Affiliations
David Thomas Hughes, Univ. of Edinburgh (United Kingdom)
S. McLaughlin, Univ. of Edinburgh (United Kingdom)
Gary Ushaw, Univ. of Edinburgh (United Kingdom)
Bernard Mulgrew, Univ. of Edinburgh (United Kingdom)
David S. Broomhead, Defence Research Agency (United Kingdom)

Published in SPIE Proceedings Vol. 2038:
Chaos in Communications
Louis M. Pecora, Editor(s)

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