For a continuously differentiable dynamical system , 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.
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In dynamical systems, an equilibrium point is a point where the system can remain indefinitely if it starts there, assuming no external disturbances. An equilibrium point is classified based on its stability properties, which are determined by analyzing the behavior of the system near that point. A **hyperbolic equilibrium point** is a specific type of equilibrium point where the linearization of the system at that point has no eigenvalues with zero real parts.