= Linearization stability theorem
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.
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