Nonlinear symmetries of plane polynomial differential equations
by
Sebastian Walcher
Zentrum Mathematik, TU München, Germany

The investigation of Lie symmetries of ordinary differential equations is important for systematic reduction and integration procedures as well as for qualitative theory. The first of these topics is classical. The second is more recent and has been studied in great detail for (linear) compact symmetry groups, yet there seems to be little work on nonlinear symmetries. Even in the case of plane ordinary differential equations, nonlinear symmetries are relevant in cases of non-generic behavior, such as existence of centers or homoclinic orbits.

In dimension two, existence of a nontrivial Lie symmetry is equivalent to existence of an integrating factor. These always exist locally near a nonstationary point, thus one has to impose more specific conditions to make the setting nontrivial. In computational problems it seems unavoidable to restrict the class of admissible functions.

In this talk we present a number of results on plane polynomial systems admitting algebraic integrating factors (equivalently, algebraic infinitesimal symmetries). The first set of results deals with the construction and enumeration of polynomial systems with prescribed algebraic integrating factors. The second set addresses the problem of deciding whether a given system admits an algebraic integrating factor, and computing it in the case of an affirmative answer. In the generic case this problem has been solved completely.

Date received: March 22, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafv-83.