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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

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