Consider an autonomous differential equation \begin{equation} \dot{x}=f(x). \label{eq} \end{equation}

An equilibrium point $x_0$ of () is such that $f(x_0)=0$ . Conversely a regular point of () 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 $x_0$ , $U'\subset U$ such that every solution of () with initial condition in $U'$ (i.e. $x(0)\in U'$ ), satisfies $$x(t)\in U$$ for all $t\geq0$ .

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 $U''$ such that every solution of () with initial condition in $U''$ (i.e. $x(0)\in U''$ ) satisfies $$\lim_{t\to\infty}x(t)=x_0.$$