 Hartman–Grobman theorem

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.

The Hartman–Grobman theorem is a result in the field of differential equations and dynamical systems, named after mathematicians Philip Hartman and Robert Grobman. The theorem provides a powerful tool for analyzing the local behavior of nonlinear dynamical systems near equilibrium points.