On the Hartman-Grobman Theorem

Date

2022-08

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Abstract

Reducing a nonlinear system x˙ = f(x) (0.1) to a simpler form by choosing correct coordinates has always been a research direction. The Hartman- Grobman Theorem showed that if f is continuously differentiable, then there is a neighborhood of a hyperbolic equilibrium point and a homeomorphism on this neighborhood, such that the system in this neighborhood is changed to a linear system under such a homeomorphism [8, 9, 10, 18]. In this dissertation, we showed that for any bounded neighborhood of a hyperbolic equilibrium point x0, there is a transformation which is locally homeomorphism, such that the system is changed into a linear system in this neighborhood. If the eigenvalues of Df(x0) are all located in the left-half (or right-half) complex plane, then there is a homeomorphism on the whole region of attraction (or repulsion) such that the nonlinear system on the region of attraction/repulsion is changed into a linear system under such a coordinate change.


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Keywords

Hartman-Grobman Theorem, Linearization, Transformation

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