Electron. J. Diff. Eqns., Vol. 2001(2001), No. 22, pp. 1-15.

Dynamics of polynomial systems at infinity

Efthimios Kappos

The behaviour of dynamics at infinity has not received much attention, even though it was central to Poincare's analysis of qualitative dynamics. Poincare's `sphere' is actually a projective plane and our treatment of dynamics at infinity in more than two dimensions requires the use of $\mathbb{R} P^n$. In control theory, `strange' transients have been reported by Kokotovic and Sussmann, where they go by the name of `peaking behaviour'. These have a simple explanation when we consider the dynamics on the Poincare compactification of state space. In this work, we propose to give an analysis of the issues arising in trying to examine the dynamics at infinite radius for dynamical systems in arbitrary dimension. Use is made of the Newton polytope and of recent results on principal parts of vector fields.

Submitted August 16, 2000. Published April 4, 2001.
Math Subject Classifications: 34C11, 34D23, 37B30, 52B12.
Key Words: Dynamics on manifolds, Newton polytopes, dissipative systems, peaking, Poincare and Bendixson spheres.

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Efthimios Kappos
Department of Applied Mathematics
University of Sheffield
Sheffield, S3 7RH, U.K.
e-mail: E.Kappos@shef.ac.uk

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