HartmanGrobman

HartmanGrobman, more commonly spelled the Hartman–Grobman theorem, is a fundamental result in dynamical systems that describes the local behavior of nonlinear systems near hyperbolic equilibria. It asserts that, near such an equilibrium, the flow of a nonlinear system is topologically conjugate to the flow of its linearization, implying that the nonlinear dynamics are qualitatively the same as those of the linearized system.

In precise terms, consider a continuously differentiable vector field f: R^n -> R^n with an equilibrium x0 (f(x0) = 0). Let A = Df(x0) be the Jacobian matrix. If A is hyperbolic (no eigenvalues with zero real part), then there exist neighborhoods of x0 and of 0 and a homeomorphism h mapping one neighborhood to the other, with h(x0) = 0, such that the local flow φ_t of the nonlinear system and the linear flow e^{tA} are conjugate on those neighborhoods: φ_t(x) = h(e^{tA}(h^{-1}(x))) for all t and x in the appropriate domain. In effect, the nonlinear dynamics near x0 replicate the qualitative features of the linear system, including the structure of stable and unstable manifolds.

A discrete-time analog exists for diffeomorphisms, stating that the local dynamics near a hyperbolic fixed point are topologically conjugate to the dynamics of the linear part. The theorem is widely used to justify linearization as a tool for local stability analysis, while also highlighting the limits when hyperbolicity fails.