Classification of Quadratic Systems Admitting the Existence of an Algebraic Limit Cycle

In the paper we find a set of necessary conditions that must be satisfied by a quadratic system in order to have an algebraic limit cycle. We find a countable set of less than 5 parameter families of quadratic systems such that every quadratic system with an algebraic limit cycle must, after a change of variables, belong to one of those families. We provide a classification of all the quadratic systems which can have an algebraic limit cycle based on geometrical properties of the embedding of the system in the Poincare compactification of R^2. We propose names for all the classes we distinguish and we classify all known examples of quadratic systems with algebraic limit cycle. We also prove the integrability of certain classes of quadratic systems.

By: Jaume Llibre, Grzegorz Swirszcz

Published in: Bulletin des Sciences Mathématiques, volume 131, (no 5), pages 405-421 in 2007

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