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[[File:Pendulum Phase Portrait.jpg|400px|thumbnail|Illustration of how a phase portrait would be constructed for the motion of a simple pendulum.]] | |||
[[File:Pendulumphase.png|right|300px|thumb|[[Potential energy]] and phase portrait of a [[simple pendulum]]. Note that the x-axis, being angular, wraps onto itself after every 2π radians.]] | |||
[[File:Van der pols equation phase portrait.jpg|right|300px|thumb|Phase portrait of [[Van der Pol oscillator|van der Pol's equation]], <math>\frac{d^2y}{dt^2}+\epsilon(y^2-1)\frac{dy}{dt}+y=0,\quad\epsilon=1</math>]] | |||
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 (graphics)|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 conjugacy|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]] | |||
* [[Complex_quadratic_polynomial#Parameter_plane|Parameter plane ( c-plane)]] and [[Mandelbrot set]] | |||
== See also == | |||
*[[Phase space]] | |||
*[[Phase plane]] | |||
*[[Phase plane method]] | |||
==References== | |||
*{{cite book | |||
| last = Jordan | |||
| first = D. W. | |||
| last2 = Smith | |||
| first2 = P. | |||
| year = 2007 | |||
| edition = fourth | |||
| title = Nonlinear Ordinary Differential Equations | |||
| publisher = Oxford University Press | |||
| isbn = 978-0-19-920824-1 | |||
}} Chapter 1. | |||
*{{cite book|author=Steven Strogatz|title=Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering|year=2001|isbn=9780738204536}} | |||
==External links== | |||
*http://economics.about.com/od/economicsglossary/g/phase.htm | |||
*http://www.enm.bris.ac.uk/staff/berndk/chaosweb/state.html | |||
[[Category:Dynamical systems]] |
Latest revision as of 10:50, 15 March 2013
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
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Chapter 1. - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534