Hartman-Grobman theorem

It can be used to prove the Hartman-Grobman theorem, which describes the qualitative behaviour of certain differential equations near certain equilibria.

Philip Hartman (born 16 May 1915, Baltimore) is an American mathematician at Johns Hopkins University working on differential equations who introduced the Hartman-Grobman theorem.

In particular, at each equilibrium of a smooth dynamical system with an n-dimensional phase space, there is a certain nxn matrix A whose eigenvalues characterize the behavior of the nearby points (Hartman-Grobman theorem).

The Hartman-Grobman theorem states that the orbit structure of a dynamical system in a neighbourhood of a hyperbolic equilibrium point is topologically equivalent to the orbit structure of the linearized dynamical system.

In mathematics, in the study of dynamical systems, the Hartman-Grobman theorem or linearization theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point.

In the hyperbolic case the Hartman-Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map J x. The hyperbolic case is also structurally stable.