Hyperbolic equilibrium point

A hyperbolic equilibrium point has no Jacobian eigenvalue with zero real part. Every linearized mode therefore has an exponentially growing or decaying direction, with no neutral center direction at the linear level. Stable nodes, stable foci and saddles can all be hyperbolic; hyperbolic does not mean attracting.

The absence of neutral eigenvalues makes local classification particularly useful. Small smooth nonlinear terms preserve the qualitative local structure described by the linearization under the hypotheses of the Hartman-Grobman theorem. For example, $\dot{x}=-x$, $\dot{y}=y$ is hyperbolic but unstable because of its growing direction. As a parameter moves an eigenvalue onto the imaginary axis, hyperbolicity fails and nonlinear terms can organize a bifurcation. This identifies where a familiar linear classification needs additional analysis.

For a continuously differentiable dynamical system $\dot{x}=f(x)$, a fixed point is hyperbolic if no eigenvalue of its Jacobian has zero real part. In a neighborhood of a hyperbolic fixed point, the nonlinear flow is locally topologically conjugate to its linearized flow. Thus the saddle, attracting or repelling character persists under the nonlinear terms. The theorem does not decide the nonlinear character of an equilibrium with purely imaginary or zero eigenvalues: a conserved Hamiltonian, Lyapunov function or higher-order analysis may be needed instead.

If every Jacobian eigenvalue has negative real part, the equilibrium is locally asymptotically stable.

More generally, if the equilibrium is hyperbolic, the Hartman-Grobman theorem transfers the local attracting, repelling or saddle behavior of the linearized vector field to the nonlinear flow. Two real eigenvalues with opposite signs give a saddle equilibrium; all real parts positive give a source. Purely imaginary or zero eigenvalues require a separate nonlinear analysis. Merely having distinct eigenvalues does not remove this exception.