Phase portrait

Potential energy and phase portrait of a simple pendulum. Note that the x-axis, being angular, wraps onto itself after every 2π radians.

Illustration of how a phase portrait would be constructed for the motion of a simple pendulum.

Phase portrait of van der Pol's equation, ${\frac {d^{2}y}{dt^{2}}}+\epsilon (y^{2}-1){\frac {dy}{dt}}+y=0,\quad \epsilon =1$

A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different curve, or point.

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.

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.

Examples

• 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 (right).
• Bifurcation diagram
• Parameter plane (c-plane) and Mandelbrot set

See also

• Phase space
• Phase plane
• Phase plane method

References

• 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.

External links

• Phase Portrait Generator a tool for sketching phase portraits of 2D systems.
• Linear Phase Portraits, an MIT Mathlet.