Hartman–Grobman theorem

In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearization theorem is a theorem about the local behavior of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearization—a natural simplification of the system—is effective in predicting qualitative patterns of behavior.

The theorem states that the behavior of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behavior of its linearization near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearization has real part equal to zero. Therefore, when dealing with such dynamical systems one can use the simpler linearization of the system to analyze its behavior around equilibria.

Consider a system evolving in time with state ${\displaystyle u(t)\in \mathbb {R} ^{n}}$ that satisfies the differential equation ${\displaystyle du/dt=f(u)}$ for some smooth map ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$. Suppose the map has a hyperbolic equilibrium state ${\displaystyle u^{*}\in \mathbb {R} ^{n}}$: that is, ${\displaystyle f(u^{*})=0}$ and the Jacobian matrix ${\displaystyle A=[\partial f_{i}/\partial x_{j}]}$ of ${\displaystyle f}$ at state ${\displaystyle u^{*}}$ has no eigenvalue with real part equal to zero. Then there exists a neighborhood ${\displaystyle N}$ of the equilibrium ${\displaystyle u^{*}}$ and a homeomorphism ${\displaystyle h:N\to \mathbb {R} ^{n}}$, such that ${\displaystyle h(u^{*})=0}$ and such that in the neighbourhood ${\displaystyle N}$ the flow of ${\displaystyle du/dt=f(u)}$ is topologically conjugate by the continuous map ${\displaystyle U=h(u)}$ to the flow of its linearization ${\displaystyle dU/dt=AU}$.

...
Wikipedia