Hartman–Grobman theorem

The set of Hartman - Grobman, also known as linearization theorem, states that the behavior of a dynamic system in the neighborhood of a hyperbolic fixed point is similar to the behavior of the at this point the linearized system.

Named the sentence after the Americans Philip Hartman and the Russians David Grobman, separated from each other in 1960 and 1959 respectively, published the sentence independently.

Set

Be

A mapping of the form

And

Here, and constant matrices with the eigenvalues ​​or for whose real parts applies:

Then there is a homeomorphism

Between a neighborhood of a neighborhood such that

With

More generally, a system can be of the form with by a linear coordinate transformation always bring in the above form if all the eigenvalues ​​of non- vanishing real part have.

Example

Be

The only fixing point of the system. Then

The Jacobian matrix at this point, with the linearizer of the system according

So

The eigenvalues ​​of,

Have real parts different from zero, thus is a hyperbolic fixed point and the conditions of the theorem of Hartman - Grobman are met. Since the eigenvalues ​​have different signs, it is a saddle point and hence an unstable fixed point. By Theorem this now applies not only to the linearized, but also for the original system.