= Hyperbolic equilibrium point
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= Hyperbolic equilibrium
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= Hyperbolic equilibria
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= Hyperbolic fixed point
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= Hyperbolic
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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.
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