Parametric qualitative analysis of ordinary differential equations: Computer algebra methods for excluding oscillations (extended abstract) (invited talk)

Abstract

Investigating oscillations for parametric ordinary differential equations (ODEs) has many applications in science and engineering but is a very hard problem. Already for two dimensional polynomial systems this question is related to Hilbert's 16th problem, which is still unsolved [1]. Using the theory of Hopf-bifurcations some non-numeric algorithmic methods have been recently developed to determine ranges of parameters for which some small stable limit cycle will occur in the system [2,3,4,5,6,7,8]. These algorithms give exact conditions for the existence of fixed points undergoing a Poincar'e Andronov-Hopf bifurcation that give birth to a small stable limit cycle under some general conditions which can be made algorithmic, too. If these conditions are not satisfied, one can be sure that there are no such fixed points, but unfortunately one cannot conclude that there are no limit cycles-which could arise by other means. Nevertheless, it is tempting to conjecture even in these cases that there are no oscillations, as has been done e.g. in [5,6]. © 2010 Springer-Verlag Berlin Heidelberg.