a hyperbolic equilibrium
Equilibria come in two moods. A 'robust' one has a clear verdict — it either pulls everything in or pushes things out, and a tiny change to the equations will not overturn that verdict. A 'borderline' one sits on a knife's edge, where the smallest nudge could tip it either way. Hyperbolic is the precise name for the robust kind, the kind whose fate the linearization can read off with confidence.
The definition is purely about eigenvalues. An equilibrium x* is hyperbolic when every eigenvalue of the Jacobian matrix J at x* has nonzero real part — none of them lies exactly on the imaginary axis. Because the real part of an eigenvalue controls whether its mode grows (positive real part) or decays (negative real part), having no zero real part means every direction is decisively either expanding or contracting. Sinks, sources, and saddles are all hyperbolic; centers (purely imaginary eigenvalues) and any equilibrium with a zero eigenvalue are not.
Hyperbolicity is the exact condition under which the Hartman-Grobman theorem applies, so it is the dividing line between equilibria you can fully understand by linearizing and those you cannot. It also brings robustness: a hyperbolic equilibrium keeps its type under small perturbations of the system, which is why hyperbolic points are 'generic' (typical) and non-hyperbolic ones are special — they are precisely where bifurcations are born.
The Jacobian [−2, 1; 0, 3] has eigenvalues −2 and 3, neither with zero real part, so this equilibrium is hyperbolic (a saddle); by contrast [0, −1; 1, 0] has eigenvalues plus and minus i, both with zero real part, so that equilibrium is non-hyperbolic.
No eigenvalue on the imaginary axis means hyperbolic; an eigenvalue with zero real part (a pure imaginary pair or a zero) makes it non-hyperbolic.
Hyperbolic does not mean 'saddle' — sinks and sources are hyperbolic too. The word refers only to having no eigenvalue with zero real part, not to the geometry of hyperbola-shaped trajectories.