On the n-Dimensional Phase Portraits

Phase Portraits of Linear Systems

1.1. One positive and one negative eigenvalue. This is described in Example 4.2.1 of [1]. Here the origin is a saddle: unstable but neither a repeller nor an attractor. Draw the two eigenlines (the lines defined by eigenvectors). Put two “outward” arrows on the eigenline that corresponds to the positive eigenvalue and two “inward” arrows on the eigenline for the negative eigenvalue. Fill in the...

Global phase portraits 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...

Phase portraits of uniform isochronous quartic centers

In this paper we classify the global phase portraits in the Poincaré disc of all quartic polynomial differential systems with a uniform isochronous center at the origin.

Phase Portraits for Quadratic Homogeneous Polynomial Vector Fields on S

Let X be a homogeneous polynomial vector field of degree 2 on S. We show that if X has at least a non–hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on S is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 Hilbert’s ...

'Phase portraits' for characterizing advanced lasers