Phase portrait

Short description: Plot of a dynamical system's trajectories in phase space

Differential equations

Navier–Stokes differential equations used to simulate airflow around an obstruction.

Scope

* Natural sciences
* Engineering

* Astronomy
* Physics
* Chemistry

Biology
* Geology

Applied mathematics

* Continuum mechanics
* Chaos theory
* Dynamical systems

Social sciences

* Economics
* Population dynamics

Classification

Types

* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear

By variable type

* Dependent and independent variables

* Autonomous
* Complex
* Coupled / Decoupled
* Exact
* Homogeneous / Nonhomogeneous

Features

* Order
* Operator

* Notation

Relation to processes

* Difference (discrete analogue)

* Stochastic
* Stochastic partial
* Delay

Solution

General topics

* Picard–Lindelöf theorem
* Wronskian

* Phase portrait
* Phase space

* Lyapunov / Asymptotic / Exponential stability
* Rate of convergence
* Series / Integral solutions

* Numerical integration
* Dirac delta function

Solution methods

* Inspection
* Method of characteristics

Euler
* Exponential response formula
* Finite difference (Crank–Nicolson)
* Finite element
* Infinite element
* Finite volume
* Galerkin
* Petrov–Galerkin
* Integrating factor
* Integral transforms
* Perturbation theory
* Runge–Kutta

* Separation of variables
* Undetermined coefficients
* Variation of parameters

People

* Isaac Newton
* Gottfried Leibniz
* Leonhard Euler
* Émile Picard
* Józef Maria Hoene-Wroński
* Ernst Lindelöf
* Rudolf Lipschitz
* Augustin-Louis Cauchy
* John Crank
* Phyllis Nicolson
* Carl David Tolmé Runge
* Martin Kutta

* v
* t
* e

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 x¨+2γx˙+ω2x=0.

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.

## Contents

* 1 Examples
* 2 Visualizing the behavior of ordinary differential equations
* 3 See also
* 4 References
* 5 External links

## Examples

Illustration of how a phase portrait would be constructed for the motion of a simple pendulum. Phase portrait of van der Pol's equation, d2ydt2+μ(y2−1)dydt+y=0.

* 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. [1]

Stability[1] Unstable  | Most of the system's solutions tend towards ∞ over time

|

Asymptotically stable  | All of the system's solutions tend to 0 over time
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 (trace = λ1 \+ λ2, determinant = λ1 x λ2) of the system.[1]

Phase Portrait Behavior[1] Eigenvalue, Trace, Determinant  | Phase Portrait Shape

|

λ1 & λ2 are real and of opposite sign;

Determinant < 0

| Saddle (unstable)
λ1 & λ2 are real and of the same sign, and λ1 ≠ λ2;

0 < determinant < (trace2 / 4)

| Node (stable if trace < 0, unstable if trace > 0)
λ1 & λ2 have both a real and imaginary component;

(trace2 / 4) < determinant

| Spiral (stable if trace < 0, unstable if trace > 0)

## See also

* Phase space
* Phase plane

## References

1. ↑ 1.0 1.1 1.2 1.3 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/18-03-differential-equations-spring-2010/resources/mit18_03s10_chapter_26/)

* 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. https://archive.org/details/nonlineardynamic00stro.

## External links

* Linear Phase Portraits, an MIT Mathlet.

* v
* t
* e

Differential equations

Classification|

| Operations|

* Differential operator
* Notation for differentiation
* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear
* Holonomic

|

Attributes of variables|

* Dependent and independent variables
* Homogeneous
* Nonhomogeneous
* Coupled
* Decoupled
* Order
* Degree
* Autonomous
* Exact differential equation
* Complex differential equation

Relation to processes|

* Difference (discrete analogue)
* Stochastic
* Stochastic partial
* Delay

Solutions|

| Solution topics|

* Picard–Lindelöf theorem (existence and uniqueness)
* Wronskian
* Phase portrait
* Phase space
* Lyapunov stability
* Asymptotic stability
* Exponential stability
* Rate of convergence
* Series solutions
* Integral solutions
* Numerical integration
* Dirac delta function

|

Solution methods|

* Inspection
* Separation of variables
* Method of undetermined coefficients
* Variation of parameters
* Integrating factor
* Integral transforms
* Euler method
* Finite difference method
* Crank–Nicolson method
* Runge–Kutta methods
* Finite element method
* Finite volume method
* Galerkin method
* Perturbation theory

Applications|

* List of named differential equations

Mathematicians|

* Isaac Newton
* Gottfried Wilhelm Leibniz
* Leonhard Euler
* Jacob Bernoulli
* Émile Picard
* Józef Maria Hoene-Wroński
* Ernst Lindelöf
* Rudolf Lipschitz
* Joseph-Louis Lagrange
* Augustin-Louis Cauchy
* John Crank
* Phyllis Nicolson
* Carl David Tolmé Runge
* Martin Kutta

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Phase portrait. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Phase_portrait&oldid=189870"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template