Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the phase space. This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value. The concept of topological equivalence is important in classifying the behaviour of systems by specifying when two different phase portraits represent the same qualitative dynamic behavior. An attractor is a stable point which is also called a "sink". The repeller is considered as an unstable point, which is also known as a "source".

A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stable steady states (with dots) and unstable steady states (with circles) in a phase space. The axes are of state variables.

Visualizing the behavior of ordinary differential equations

A phase portrait represents the directional behavior of a system of ordinary differential equations (ODEs). The phase portrait can indicate the stability of the system. ^{[1]}

Stability^{[1]}

Unstable

Most of the system's solutions tend towards ∞ over time

Asymptotically stable

All of the system's solutions tend to 0 over time

Neutrally stable

None of the system's solutions tend towards ∞ over time, but most solutions do not tend towards 0 either

The phase portrait behavior of a system of ODEs can be determined by the eigenvalues or the trace and determinant (trace = λ_{1} + λ_{2}, determinant = λ_{1} x λ_{2}) of the system.^{[1]}

Phase Portrait Behavior^{[1]}

Eigenvalue, Trace, Determinant

Phase Portrait Shape

λ_{1} & λ_{2} are real and of opposite sign;

Determinant < 0

Saddle (unstable)

λ_{1} & λ_{2} are real and of the same sign, and λ_{1} ≠ λ_{2};

0 < determinant < (trace^{2} / 4)

Node (stable if trace < 0, unstable if trace > 0)

λ_{1} & λ_{2} have both a real and imaginary component;

(trace^{2} / 4) < determinant

Spiral (stable if trace < 0, unstable if trace > 0)

^ ^{a}^{b}^{c}^{d}Haynes Miller, and Arthur Mattuck. 18.03 Differential Equations. Spring 2010. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA. (Supplementary Notes 26 by Haynes Miller: https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/resources/mit18_03s10_chapter_26/)

Jordan, D. W.; Smith, P. (2007). Nonlinear Ordinary Differential Equations (fourth ed.). Oxford University Press. ISBN978-0-19-920824-1. Chapter 1.