Probability of Occurrence of Some Planar Random Quasi-homogeneous Vector Fields

The objective of this work is the study of the probability of occurrence of phase portraits in a family of planar quasi-homogeneous vector fields of quasi degree q, that is a natural extension of planar linear vector fields, which correspond to q = 1. We obtain the exact values of the corresponding probabilities in terms of a simple one-variable definite integral that only depends on q. This integral is explicitly computable in the linear case, recovering known results, and it can be expressed in terms of either complete elliptic integrals or of generalized hypergeometric functions in the non-linear one. Moreover, it appears a remarkable phenomenon when q is even: the probability to have a center is positive, in contrast with what happens in the linear case, or also when q is odd, where this probability is zero.