equilibrium point
Consider an autonomous differential equation
(1) |
An equilibrium point of (1) is such that . Conversely a regular point of (1) is such that .
If the linearization has no eigenvalue with zero realpart, is said to be a hyperbolic equilibrium, whereas ifthere exists an eigenvalue with zero real part, the equilibriumpoint is nonhyperbolic.
An equilibrium point is said to be stable if forevery neighborhood , there exists a neighborhoodof , such that every solution of (1)with initial condition in (i.e. ),satisfies
for all .
Consequently an equilibrium point is said to beunstable if it is not stable.
Moreover an equilibrium point is said to beasymptotically stable if it is stable and there exists such that every solution of (1) with initialcondition in (i.e. ) satisfies
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 |