Hartman–Grobman theorem

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

Basically the theorem states that the behaviour of a dynamical system near a hyperbolic fixed point is qualitatively the same as the behaviour of its linearization near the origin. Therefore when dealing with such fixed points we can use the simpler linearization of the system to analyze its behaviour.

Hartman-Grobman theorem

Let

$f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$

be a smooth map with a hyperbolic fixed point p. Let A denote the linearization of f at point p. Then there exists a neighborhood U of p and a homeomorphism

$h:U\to \mathbb {R} ^{n}$

such that

$f_{U}=h^{-1}\circ A\circ h;$

that is, in a neighbourhood U of p, f is topologically conjugate to its linearization. Note that this only applies if the eigenvalues of the linearization A all have absolute value different from one. In general, even for infinitely differentiable maps $f$, the homomorphism $h$ need not to be smooth, nor even locally Lipschitz. However, it turns out to be Hölder continuous, with an exponent depending on the constant of hyperbolicity of A.

External links

• Coayla-Teran, E. (2007). "Hartman-Grobman Theorems along Hyperbolic Stationary Trajectories" (PDF). Discrete and Continuous Dynamical Systems. 17 (2): 281–292. doi:10.3934/dcds.2007.17.281. Retrieved 2007-03-09. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)