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March  2008, 1(1): 15-25. doi: 10.3934/dcdss.2008.1.15

Counting uniformly attracting solutions of nonautonomous differential equations

Arno Berger 1,

1. 

Department of Mathematics and Statistics, University of Canterbury, Christchurch

Received  September 2006 Revised  August 2007 Published  December 2007

Bounded uniform attractors and repellors are the natural nonautonomous analogues of autonomous stable and unstable equilibria. Unlike for equilibria, it is generally a difficult dynamical task to determine the number of uniformly attracting or repelling solutions for a given nonautonomous equation, even if the latter exhibits strong structural properties such as e.g. polynomial growth in space or periodicity in time. The present note highlights this aspect by proving that the number of uniform attractors is locally finite for several classes of equations, and by providing examples for which this number can be any $N\in \N$. These results and examples extend and complement recent work on nonautonomous differential equations.
Keywords: Poincaré map., polynomial differential equation, Nonautonomous dynamical system, attractor, repellor.
Mathematics Subject Classification: Primary: 34A12, 37C70; Secondary: 34C0.
Citation: Arno Berger. Counting uniformly attracting solutions of nonautonomous differential equations. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 15-25. doi: 10.3934/dcdss.2008.1.15
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