Compactified dynamics and peaking

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 RPⁿ. Perhaps because many practical systems are `dissipative´, attention has focused on `local´ problems where the theory of normal forms plays a major role. In control theory, however, `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