# Phase portrait

Differential equations :
Scope :
Fields
Natural sciencesEngineering
Astronomy
Physics
Chemistry
Biology
Geology
Applied mathematics
Continuum mechanics
Chaos theory
Dynamical systems
Social sciences
Economics
Population dynamics
List of named differential equations :
Classification :
Types
Ordinary
Partial
Differential-algebraic
Integro-differential
Fractional
Linear
Non-linear
By variable type
Dependent and independent variables
Autonomous
Coupled / Decoupled
Exact
Homogeneous / Nonhomogeneous
Features
Order
Operator
Notation :
Relation to processes
Difference (discrete analogue)
Stochastic
Stochastic partial
Delay :
Solution :
Existence and uniqueness
Picard–Lindelöf theorem
Peano existence theorem
Carathéodory's existence theorem
Cauchy–Kowalevski theorem :
General topics
Initial conditions
Boundary values
Dirichlet
Neumann
Robin
Cauchy problem
Wronskian
Phase portrait
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
Green's function
Integrating factor
Integral transforms
Perturbation theory
Runge–Kutta
Separation of variables
Undetermined coefficients
Variation of parameters :
People :
List
Isaac Newton
Gottfried Leibniz
Jacob Bernoulli
Leonhard Euler
Joseph-Louis Lagrange
Józef Maria Hoene-Wroński
Joseph Fourier
Augustin-Louis Cauchy
George Green
Carl David Tolmé Runge
Martin Kutta
Rudolf Lipschitz
Ernst Lindelöf
Émile Picard
Phyllis Nicolson
John Crank :
vte :
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.
Stability :
| 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 = λ$_1$ + λ$_2$, determinant = λ$_1$ x λ$_2$) of the system.
Phase Portrait Behavior :
| Eigenvalue, Trace, Determinant | Phase Portrait Shape |
| λ$_1$ & λ$_2$ are real and of opposite sign; Determinant < 0 | Saddle (unstable) |
| λ$_1$ & λ$_2$ are real and of the same sign, and λ$_1$ ≠ λ$_2$; 0 < determinant < (trace$^2$ / 4) | Node (stable if trace < 0, unstable if trace > 0) |
| λ$_1$ & λ$_2$ have both a real and imaginary component; (trace$^2$ / 4) < determinant | Spiral (stable if trace < 0, unstable if trace > 0) |