Qualitative analysis of phase--portrait for a class of planar vector fields via the comparison method

Phase portraits of random planar homogeneous vector fields

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.

二次与三次平面拟齐次微分系统的全局动力学

Global Dynamics Of Quadratic And Cubic Planar Quasi-homogeneous Differential Systems

本文研究了二次与三次平面拟齐次非齐次微分系统的全局动力学，通过将其化为齐次系统并运用blow-up等方法，刻画了系统的全局相图。

Global phase portrait of a SIS model

In the qualitative theory of ordinary differential equations, we can find many papers whose objective is the classification of all the possible topological phase portraits of a given family of differential system. Most of the studies rely on systems with real parameters and the study consists of outlining their phase portraits by finding out some conditions on the parameters. Here, we studied a susceptible-infected-susceptible (SIS) model described by the differential system $\dot{x}=-bxy-mx+cy+mk$, $\dot{y}=bxy-(m+c)y$, where $b$, $c$, $k$, $m$ are real parameters with $b \neq 0$, $m \neq 0$ …

Global Phase Portrait and Large Degree Asymptotics for the Kissing Polynomials

We study a family of monic orthogonal polynomials which are orthogonal with respect to the varying, complex valued weight function, $\exp(nsz)$, over the interval $[-1,1]$, where $s\in\mathbb{C}$ is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is $s\in i \mathbb{R}$, due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as $n\to\infty$ have been recently studied for $s\in i\mathbb{R}$, and our main goal is to extend these results to all $s$…