Thursday, August 25, 2016

The Hartman-Grobman Linearization Theorem

Theorem: In the neighborhood of a hyperbolic fixed point, a smooth vector field or a diffeomorphism is topologically conjugate to its linear part.

This result was proved by Grobman and Hartman independently around 1959-1960 and basically states that the dynamics near a hyperbolic fixed point is essentially the same as the dynamics of its linearization which we can characterize completely from the eigenvalues pattern.  This is true for both continuous-time dynamics (vector field) or discrete-time dynamics (diffeomorphism).

Here is a sketch of the standard proof for the case of  a diffeomorphism.  First, we need the following simple fact for linear maps in Banach spaces: if $F$ is an invertible contraction, then $I+F^{-1}$ is also invertible. This can be seen as follows. $I+F^{-1} = F^{-1}(I+F)$.  If $I+F$ is not invertible, then there exists $x\neq y$ such that $x+F(x) = y+F(y)$.  This implies that $x-y = F(y)-F(x)$, i.e. $\|x-y\| = \|F(y)-F(x)\|$, contradicting the fact that $F$ is a contraction. Therefore $I+F$ is invertible, and thus $I+F^{-1}$ is invertible since it is the product of two invertible maps.

Consider a diffeomorphism $f$ with a hyperbolic fixed point at $0$. Let $A$ be the linear part of $f$ at $0$.  We want to find a homeomorphism $h = I+\delta$ such that $fh = hA$.  As we are interested only at $f$ near a neighborhood of $0$, we can assume that $f$ can be written as $f = A+\phi_1$ such that $\phi_1$ is bounded and have a small Lipschitz constant.  Furthermore, $\phi_1$ can be chosen small enough such that $A+\phi_1$ is a homeomorphism. Consider the equation $(A+\phi_1)h = h(A+\phi_2)$.  After using the fact that $h=I+\delta$ and some manipulation, we get the following Eq. (1):
$\delta - A^{-1}\delta(A+\phi_2) = A^{-1}(\phi_2-\phi_1(I+\delta))$
Next we argue that the linear operator $H: \delta \rightarrow \delta - A^{-1} \delta(A+\phi_2)$ is invertible.
By hyperbolicity of $A$, we can decompose the phase space into the stable subspace $W^s$ and the unstable subspace $W^u$. Since $W^s$ and $W^u$ are invariant under $A^{-1}$, if $\delta$ is a bounded function into $W^s$ and $W^u$ then $H(\delta)$ is also a bounded function into $W^s$ and $W^u$ respectively. Split $\delta = \delta^s + \delta^u$ into two functions $\delta^s$ and $\delta^u$ which maps into $W^s$ and $W^u$ respectively.
The map  $\delta^s \rightarrow A^{-1}\delta^s(A+\phi_1)$ is invertible with inverse $\delta^s \rightarrow A \delta^s (A+\phi_1)^{-1}$ since  $A \delta^s (A+\phi_1)^{-1} = A^s\delta^s(A+\phi_1)^{-1}$ the map
$\delta^s \rightarrow A \delta^s (A+\phi_1)^{-1}$ is a contraction and therefore
the map $\delta^s \rightarrow \delta^s - A^{-1}\delta^s(A+\phi_1)$ is invertible based on the fact discussed before. The same thing can be done with the $W^u$ and this implies that $H$ is invertible.

Coming back to Eq. (1) above, we get
$\delta = H^{-1}A^{-1}(\phi_2-\phi_1(I+\delta)) = \psi(\delta)$

For small $\phi_1$ and $\phi_2$, $\psi$ is a contraction and thus for given $\phi_1$ and $\phi_2$ there exists a unique $\delta$ and hence a unique $h$.  It can be shown that $h$ is a homeomorphism and by choosing $\phi_2 = 0$ we get the desired result.

References
D. M. Grobman, "Homeomorphisms of systems of differential equations," Doklady Akademii Nauk SSSR, vol. 128, pp. 880–881, 1959.
P.  Hartman, "A lemma in the theory of structural stability of differential equations," Proc. AMS, vol. 11, no. 4, pp. 610–620, 1960.