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 $f(x_0)\ne 0$.

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. $x\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{\in }{U}^{\mathrm{\prime }}$), satisfies

$$x(t)\in U$$

for all $t\ge 0$.

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. $x\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{\in }{U}^{\mathrm{\prime \prime }}$) satisfies

$$\underset{t\to \mathrm{\infty }}{lim}x(t)=x_0.$$

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