Hartman-Grobman theorem

From Wikipedia, the free encyclopedia

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.

[edit] 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.^[1]^[2]^[3]

[edit] References

^ Grobman, D.M. (1959). "Homeomorphisms of systems of differential equations". Dokl. Akad. Nauk SSSR 128: 880-881.
^ Hartman, Philip (August 1960). "A lemma in the theory of structural stability of differential equations". Proc. A.M.S. 11 (4): 610-620. DOI:10.2307/2034720. Retrieved on 2007-03-09.
^ Hartman, Philip (1960). "On local homeomorphisms of Euclidean spaces". Bol. Soc. Math. Mexicana 5: 220-241.

[edit] External Links

Coayla-Teran, E.; Mohammed, S. and Ruffino, P. (February 2007). "Hartman-Grobman Theorems along Hyperbolic Stationary Trajectories". Discrete and Continuous Dynamical Systems 17 (2): 281-292. Retrieved on 2007-03-09.