Area contraction in the presence of first integrals and almost global convergence

Dirk Aeyels; Filip De Smet; Bavo Langerock

doi:10.3934/dcds.2007.18.135

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2007, Volume 18, Issue 1: 135-157. Doi: 10.3934/dcds.2007.18.135

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Area contraction in the presence of first integrals and almost global convergence

1.
SYSTeMS Research Group, Dept. of Electrical Energy, Systems and Automation, Ghent University, Technologiepark-Zwijnaarde 914, 9052 Zwijnaarde

Received: March 31, 2006

Revised: October 31, 2006

Published: December 2006

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Abstract

We investigate the evolution of the area of multi-dimensional surfaces along the flow of a dynamical system with known first integrals, and we formulate sufficient conditions for area contraction.
These results, together with known results about the Hausdorff dimension and the box-counting dimension of invariant sets, are applied to systems exhibiting almost global convergence/asymptotic stability. This leads to a generalization of a well-known result on almost global convergence of a system, based on the use of density functions. We conclude with an example.

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Mathematics Subject Classification: 34D23, 37C10.

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