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单词 EquilibriumPoint
释义

equilibrium point


Consider an autonomous differential equation

x˙=f(x).(1)

An equilibrium point x0 of (1) is such that f(x0)=0. Conversely a regular point of (1) is such that f(x0)0.

If the linearization Df(x0) has no eigenvalue with zero realpart, x0 is said to be a hyperbolic equilibrium, whereas ifthere exists an eigenvalue with zero real part, the equilibriumpoint is nonhyperbolic.

An equilibrium point x0 is said to be stable if forevery neighborhood x0,U there exists a neighborhoodof x0, UU such that every solution of (1)with initial conditionMathworldPlanetmath in U (i.e. x(0)U),satisfies

x(t)U

for all t0.

Consequently an equilibrium point x0 is said to beunstable if it is not stable.

Moreover an equilibrium point x0 is said to beasymptotically stable if it is stable and there existsU′′ such that every solution of (1) with initialcondition in U′′ (i.e. x(0)U′′) satisfies

limtx(t)=x0.
Titleequilibrium point
Canonical nameEquilibriumPoint
Date of creation2013-03-22 13:18:34
Last modified on2013-03-22 13:18:34
OwnerDaume (40)
Last modified byDaume (40)
Numerical id10
AuthorDaume (40)
Entry typeDefinition
Classificationmsc 34C99
Synonymsteady state solution
Synonymfixed pointMathworldPlanetmathPlanetmath
Synonymsingular point
Defineshyperbolic equilibrium
Definesnonhyperbolic equilibrium
Definesstable
Definesunstable
Definesasymptotically stable
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