Hyperbolic equilibrium point

Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be continuously differentiable, and let $x^* \in U$ be an equilibrium point of $x' = f(x)$. The point $x^*$ is hyperbolic if no eigenvalue of the Jacobian matrix $Df(x^*)$ has real part $0$.

Remarks

A saddle, a node and a spiral of a planar linear system are hyperbolic. A centre is not. For the pendulum $x' = y$, $y' = -\sin x$, the equilibrium $(\pi, 0)$ is hyperbolic, with eigenvalues $\pm 1$. The equilibrium $(0, 0)$ is not, with eigenvalues $\pm i$. Near a hyperbolic equilibrium, the linearisation describes the system qualitatively, by the Hartman–Grobman theorem. If all eigenvalues have negative real part, the point is a sink. If all have positive real part, it is a source. Otherwise it is a saddle.

Explanations

L1 human

At a hyperbolic equilibrium, each direction of the linearisation either contracts or expands. No direction is neutral, so small changes of the field do not change the type of the equilibrium.