# Equilibrium point

﻿
Equilibrium point

In mathematics, the point $ilde mathbf\left\{x\right\}in mathbb\left\{R\right\}^n$ is an equilibrium point for the differential equation

:$frac\left\{dmathbf\left\{x\left\{dt\right\} = mathbf\left\{f\right\}\left(t,mathbf\left\{x\right\}\right)$

if $mathbf\left\{f\right\}\left(t, ildemathbf\left\{x\right\}\right)=0$ for all $t,!$.

Similarly, the point $ilde mathbf\left\{x\right\}in mathbb\left\{R\right\}^n$ is an equilibrium point (or fixed point) for the difference equation

:$mathbf\left\{x\right\}_\left\{k+1\right\} = mathbf\left\{f\right\}\left(k,mathbf\left\{x\right\}_k\right)$

if $mathbf\left\{f\right\}\left(k, ildemathbf\left\{x\right\}\right)= ildemathbf\left\{x\right\}$ for $k=0,1,2,ldots$.

Equilibria can be classified by looking at the signs of the eigenvalues of the linearization of the equations about the equilibria. That is to say, by evaluating the Jacobian matrix at each of the equilibrium points of the system, and then finding the resulting eigenvalues, the equilibria can be categorized. Then the behavior of the system in the neighborhood of each equilibrium point can be qualitatively determined, (or even quantitatively determined, in some instances, by finding the eigenvector(s) associated with each eigenvalue).

An equilibrium point is "hyperbolic" if none of the eigenvalues have zero real part. If all eigenvalues have negative real part, the equilibrium is a stable node. If all have positive real part, the equilibrium is an unstable node. If at least one eigenvalue has negative real part and at least one has positive real part, the equilibrium is a saddle point.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• equilibrium point — pusiausvyros taškas statusas T sritis automatika atitikmenys: angl. equilibrium point vok. Gleichgewichtspunkt, m rus. точка равновесия, f pranc. point d équilibre, m …   Automatikos terminų žodynas

• equilibrium point — pusiausvyros taškas statusas T sritis fizika atitikmenys: angl. equilibrium point vok. Gleichgewichtspunkt, m rus. точка равновесия, f pranc. point d’équilibre, m …   Fizikos terminų žodynas

• Hyperbolic equilibrium point — In mathematics, especially in the study of dynamical system, a hyperbolic equilibrium point or hyperbolic fixed point is a special type of fixed point.The Hartman Grobman theorem states that the orbit structure of a dynamical system in the… …   Wikipedia

• Equilibrium — is the condition of a system in which competing influences are balanced and it may refer to:cienceBiology* Equilibrioception, the sense of balance present in humans and animals * Homeostasis, the ability of an open system, especially living… …   Wikipedia

• equilibrium — A condition in a *market at which *supply and *demand are in harmony. An equilibrium point establishes a *price for a product or service, and it can be achieved in both the *short and *long terms …   Auditor's dictionary

• point d'équilibre — pusiausvyros taškas statusas T sritis automatika atitikmenys: angl. equilibrium point vok. Gleichgewichtspunkt, m rus. точка равновесия, f pranc. point d équilibre, m …   Automatikos terminų žodynas

• point d’équilibre — pusiausvyros taškas statusas T sritis fizika atitikmenys: angl. equilibrium point vok. Gleichgewichtspunkt, m rus. точка равновесия, f pranc. point d’équilibre, m …   Fizikos terminų žodynas

• Equilibrium fractionation — Equilibrium isotope fractionation is the partial separation of isotopes between two or more substances in chemical equilibrium. Equilibrium fractionation is strongest at low temperatures, and (along with kinetic isotope effects) forms the basis… …   Wikipedia

• point of equilibrium — point of balance, center of motion …   English contemporary dictionary

• Point of equilibrium (genitals) — In foreskin restoration, the point of equilibrium (or POE) is a point (actually a line around the penis) where tension toward the glans and toward the base is equal when T Tape draws the skin forward. Finding the POE One physically holds the skin …   Wikipedia