Phase portraits of quadratic systems in the class m_f=3

Abstract

The tendency that increase of the finite multiplicity from m_f=0 and 1 to m_f=2, causing an increase of the number of parameters in the system, results in a drastic increase of the number of phase portraits and continues to prevail when m_f=3 is considered. Almost a doubling occurs compared with m_f=2. Moreover, not all phase portraits are known, due to the fact that the limit cycle problem has only partly been solved for m_f=3. This effectively means that, although all possible separatrix structures are known, it is not known in all cases what the limit cycle distribution is. Prom this results that the statement that a quadratic system with m_f=3 has at most three limit cycles is still an unproved conjecture. It is known for a system in m_f=3 that, if there exist two nests of limit cycles, then there exists in each nest precisely one (hyperbolic) limit cycle [99,ZK], this verifies the above conjecture and means we still have to show that, if precisely one nest of limit cycles exists, it contains at most three limit cycles.