HartmanGrobmanTheorem

The Hartman–Grobman theorem is a fundamental result in dynamical systems that describes how nonlinear systems behave near a hyperbolic fixed point. It asserts that the local dynamics of a smooth autonomous system are qualitatively the same as those of its linearization.

Consider an autonomous ordinary differential equation x' = f(x) with f defined and continuously differentiable in a neighborhood of a fixed point x*, where f(x*) = 0. Let A = Df(x*) be the Jacobian matrix at x*. If x* is hyperbolic, meaning no eigenvalue of A has zero real part, then the nonlinear flow near x* is locally topologically conjugate to the flow of the linear system y' = Ay.

More precisely, there exist neighborhoods U of x* and V of 0, and a homeomorphism h: U → V with h(x*) = 0, such that for all t in R and x in U, h(φ_t(x)) = e^{tA} h(x), where φ_t is the nonlinear flow of f and e^{tA} is the linear flow generated by Ay. Thus the nonlinear dynamics near x* are qualitatively equivalent to the linear dynamics given by A.

Consequences include the preservation of stability type: if all eigenvalues of A have negative real parts, x* is locally asymptotically stable; if any eigenvalue has positive real part, x* is locally unstable. The theorem provides a local, not global, conjugacy and the conjugacy is generally only a homeomorphism, not necessarily smooth. The result originated in 1960 with independent work by Hartman and Grobman.