Phase Portraits of Quadratic Systems. Higher Order Singularities and Separatrix Cycles

摘要

In the thesis phase portraits of quadratic systems of differential equations in the plane are studied. First a classification is given of all phase portraits with a higher order singular point with two zero eigenvalues. Then a classification is given of all phase portraits with a third or fourth order singular point with one zero eigenvalue. In all the quadratic systems with a higher order singular point considered in the thesis, there can be at most one limit cycle. In the second part of the thesis a survey is given of all types of bounded separatrix cycles occurring in quadratic systems. For each of the five different types of these cycles the following three questions are discussed: How many limit cycles can be surrounded by a separatrix cycle; What type of foci can be situated inside a separatrix cycle; and For what conditions is the separatrix cycle stable/unstable.