Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the state 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 'sink'. The repellor is considered as an unstable point, which is also known as '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 state space. The axes are of state variables.
- Simple pendulum, see picture (right).
- Simple harmonic oscillator where the phase portrait is made up of ellipses centred at the origin, which is a fixed point.
- Van der Pol oscillator see picture (bottom right).
- Parameter plane (c-plane) and Mandelbrot set
- Jordan, D. W.; Smith, P. (2007). Nonlinear Ordinary Differential Equations (fourth ed.). Oxford University Press. ISBN 978-0-19-920824-1. Chapter 1.
- Steven Strogatz (2001). Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering. ISBN 9780738204536.
- Linear Phase Portraits, an MIT Mathlet.