A phase portrait is a

geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

representation of the trajectories of a

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

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 A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the to ...

. This reveals information such as whether an

attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...

, a

repellor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...

limit cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...

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 repeller 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.

Examples

Simple pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...

, see picture (right). *Simple

harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive const ...

where the phase portrait is made up of ellipses centred at the origin, which is a fixed point. *

Van der Pol oscillator In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation: :-\mu(1-x^2)+x= 0, where ''x'' is the position coordinate—which is a f ...

see picture (bottom right). * Parameter plane (c-plane) and

Mandelbrot set The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This ...

Visualizing the behavior of ordinary differential equations

A phase portrait represents the directional behavior of a system of

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

s (ODEs). The phase portrait can indicate the stability of the system. Haynes Miller, and Arthur Mattuck. ''18.03 Differential Equations.'' Spring 2010. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA. (Supplementary Notes 26 by Haynes Miller: https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/readings/supp_notes/MIT18_03S10_chapter_26.pdf) The phase portrait behavior of a system of ODEs can be determined by the

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...

s or the trace and

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...

(trace = λ₁ + λ₂, determinant = λ₁ x λ₂) of the system.

References

* Chapter 1. *

External links

Linear Phase Portraits
an MIT Mathlet. {{Differential equations topics Dynamical systems Diagrams

Examples

Visualizing the behavior of ordinary differential equations

See also

References

External links