Phase portrait

Phase portraits are an invaluable tool in studying dynamical systems.

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.

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.

A phase portrait represents the directional behavior of a system of ordinary differential equations (ODEs).

Potential energy and phase portrait of a simple pendulum . Note that the x-axis, being angular, wraps onto itself after every 2π radians.
Phase portrait of damped oscillator, with increasing damping strength. The equation of motion is
Illustration of how a phase portrait would be constructed for the motion of a simple pendulum.
Phase portrait of van der Pol's equation , .