# 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 |