Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 22, pp. 1-15.

Title: Dynamics of polynomial systems at infinity
        
Author: Efthimios Kappos (Univ. of Sheffield, U.K.)
         
Abstract: 
 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.