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Dynamical systems: stability and simulability

Published online by Cambridge University Press:  01 April 2007

MATHIEU HOYRUP
MATHIEU HOYRUP*
Affiliation:
LIENS, CNRS – ENS, 45 rue d'Ulm, F-75230 Paris cedex 05, France Email: hoyrup@di.ens.fr
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Computers are used extensively to simulate continuous dynamical systems. However, different conceptual and mathematical structures underlie discrete machines and continuous dynamics, so the question arises as to the ability of the computer to simulate or, more generally, to check the properties of a continuous system.

We discuss and compare two notions of stability for a continuous dynamical system, viz. shadowing and robustness, and relate them to both the practical and theoretical computability of the system. We first discuss what we can learn from the stability of a system, using a finite-precision machine. We then show, following the work in Collins (2005), that shadowing fails but robustness succeeds in ensuring the checkability of a reachability property.

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 17 , Issue 2 , April 2007 , pp. 247 - 259
Copyright
Copyright © Cambridge University Press 2007

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