Phase portraits of random planar homogeneous vector fields

TL;DR

This paper analyzes the probability distribution of phase portraits in random planar homogeneous vector fields of degrees 1 to 3, providing exact probabilities or estimates, and characterizes most portraits by their index and invariant lines.

Contribution

It offers a complete classification and probability analysis of phase portraits for degrees 1 to 3, including exact values and Monte Carlo estimates.

Findings

Most phase portraits are characterized by their index and invariant lines.

Exact probabilities are provided for degrees 1 and 2, with estimates for degree 3.

The study advances understanding of the typical behavior of random homogeneous vector fields.

Abstract

We study the phase portraits with positive probability of random planar homogeneous vector fields of degree n. In particular, for n=1,2,3, we give a complete solution of the problem and, moreover, either we give the exact value of each probability or we estimate it by using the Monte Carlo method. It is remarkable that all but two of these phase portraits are characterized by their index at the origin and by their number of invariant straight lines through it.