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Phase portrait
Plot of a dynamical system's trajectories in phase space
Plot of a dynamical system's trajectories in phase space
In mathematics, a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions is represented by a different point or curve.
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.
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.
- Damped harmonic motion, see animation (right).
- Van der Pol oscillator see picture (bottom right).
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.
| 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 (, ) of the system.
| Eigenvalue, trace, determinant | Phase portrait shape |
|---|---|
| λ1 and λ2 are real and of opposite sign; | |
| Determinant | Saddle (unstable) |
| λ1 and λ2 are real and of the same sign, and λ1 ≠ λ2; | |
| 0 2 / 4) | Node (stable if trace 0) |
| λ1 and λ2 have both a real and imaginary component; | |
| (trace2 / 4) | Spiral (stable if trace 0) |
References
- {{cite book
References
- Miller, Haynes. (Spring 2010). "Supplementary Notes to Chapter 26".
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