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

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January 2007, 18(1): 135-157. doi: 10.3934/dcds.2007.18.135

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

Dirk Aeyels ^1, , Filip De Smet ^1, and Bavo Langerock ^1,

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

Received April 2006 Revised November 2006 Published February 2007

Abstract
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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.

Keywords: Hausdorff dimension, $k$-contracting vector fields, first integrals, almost global convergence..

Mathematics Subject Classification: 34D23, 37C1.

Citation: Dirk Aeyels, Filip De Smet, Bavo Langerock. Area contraction in the presence of first integrals and almost global convergence. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 135-157. doi: 10.3934/dcds.2007.18.135

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