equilibrium point

Consider an autonomous differential equation

 $\dot{x}=f(x).$ (1)

An equilibrium point $x_{0}$ of (1) is such that $f(x_{0})=0$. Conversely a regular point of (1) is such that $f(x_{0})\neq 0$.

If the linearization $Df(x_{0})$ has no eigenvalue with zero real part, $x_{0}$ is said to be a hyperbolic equilibrium, whereas if there exists an eigenvalue with zero real part, the equilibrium point is nonhyperbolic.

An equilibrium point $x_{0}$ is said to be stable if for every neighborhood $x_{0}$,$U$ there exists a neighborhood of $x_{0}$, $U^{\prime}\subset U$ such that every solution of (1) with initial condition in $U^{\prime}$ (i.e. $x(0)\in U^{\prime}$), satisfies

 $x(t)\in U$

for all $t\geq 0$.

Consequently an equilibrium point $x_{0}$ is said to be unstable if it is not stable.

Moreover an equilibrium point $x_{0}$ is said to be asymptotically stable if it is stable and there exists $U^{\prime\prime}$ such that every solution of (1) with initial condition in $U^{\prime\prime}$ (i.e. $x(0)\in U^{\prime\prime}$) satisfies

 $\lim_{t\to\infty}x(t)=x_{0}.$
 Title equilibrium point Canonical name EquilibriumPoint Date of creation 2013-03-22 13:18:34 Last modified on 2013-03-22 13:18:34 Owner Daume (40) Last modified by Daume (40) Numerical id 10 Author Daume (40) Entry type Definition Classification msc 34C99 Synonym steady state solution Synonym fixed point Synonym singular point Defines hyperbolic equilibrium Defines nonhyperbolic equilibrium Defines stable Defines unstable Defines asymptotically stable