Hartman-Grobman and Hyperbolicity

Technique: real Jordan/spectral decomposition, then direct estimation of the matrix exponential on each Jordan block. Since $A$ is a real matrix, its complex eigenvalues occur in conjugate pairs, and there is a real invertible matrix $P$ putting $A$ into real Jordan canonical form $J = P^{-1}AP$, a block-diagonal matrix whose blocks correspond to real eigenvalues (ordinary Jordan blocks) or complex-conjugate pairs $\lambda = a\pm bi$ (real $2\times2$ Jordan blocks with $\begin{pmatrix}a&-b\\ b&a\end{pmatrix}$ on the diagonal and identity or zero blocks linking them). Group the blocks: let $J_s$ collect blocks with $\operatorname{Re}\lambda<0$ and $J_u$ collect blocks with $\operatorname{Re}\lambda>0$; since $x^*$ is hyperbolic there are no blocks with $\operatorname{Re}\lambda=0$, so this exhausts all blocks. Define $E^s = P(\text{span of coordinates of } J_s\text{-blocks})$ and $E^u$ analogously. Because $J$ is block diagonal, $J_s$- and $J_u$-coordinate subspaces are each invariant under $J$, hence (conjugating back by $P$) $E^s,E^u$ are invariant under $A$, and $\mathbb{R}^n = E^s\oplus E^u$ since the Jordan blocks partition a basis of $\mathbb{R}^n$. It remains to prove the exponential bounds. For a single Jordan block $J_k$ of size $k$ with real eigenvalue $\lambda$, we have $J_k = \lambda I + N$ where $N$ is nilpotent with $N^k=0$ and $N$ commutes with $\lambda I$, so

Hence for $t\ge0$,

where $q(t)$ is a polynomial in $t$ with positive coefficients. Choose $\alpha$ with $0<\alpha<-\lambda$ (possible since $\lambda<0$). Since $e^{-(\lambda+\alpha)t}\,\cdot(\text{const})$ dominates any fixed polynomial as $t\to\infty$ when $-(\lambda+\alpha)<0$ — indeed $\lambda+\alpha<0$ means $e^{(\lambda+\alpha)t}\to 0$ faster than $q(t)\to\infty$ — the function $t\mapsto e^{(\lambda+\alpha)t}q(t)$ is continuous on $[0,\infty)$ and tends to $0$ as $t\to\infty$, hence it is bounded by some $C_k\ge1$ on $[0,\infty)$. Therefore

The same argument with $\lambda = a\pm bi$ complex applies verbatim to the real $2\times2$-block version, since $\|e^{\begin{pmatrix}a&-b\\b&a\end{pmatrix}t}\| = e^{at}$ exactly (rotation matrices are orthogonal, hence norm $1$), and $e^{at}$ satisfies the same bound with $\alpha < -a$. Taking $C = \max_k C_k$ over the finitely many blocks composing $J_s$ (a finite maximum, since there are finitely many blocks) and $\alpha = \min_k \alpha_k$ over these blocks (again a finite minimum, hence positive), block-diagonality gives, for $v\in E^s$ written in Jordan coordinates $v = \bigoplus v_k$,

so $\|e^{J_st}v\|\le Ce^{-\alpha t}\|v\|$. Converting back through $P$ changes constants but not the exponential rates (as $P$ is a fixed invertible matrix, $\|P^{-1}\|^{-1}\|w\|\le \|Pw\|\le\|P\|\|w\|$), giving the stated bound on $E^s$ for $A$ itself, possibly after enlarging $C$. The unstable estimate for $t\le0$ on $E^u$ follows identically by applying the same argument to $-A$ restricted to $E^u$ run forward in time, i.e. replacing $t$ by $-t$ and $\lambda$ by $-\lambda>0$ so that $-\lambda<0$ plays the role of the stable case. $\blacksquare$

Proof sketch of Hartman–Grobman (technique: Lyapunov–Perron / graph transform fixed-point argument). Write $f(x^*+y) = Ay + g(y)$ where $A=Df(x^*)$ and $g(y) = o(\|y\|)$ as $y\to0$ (by definition of the derivative, since $f\in C^1$), so $g(0)=0$, $Dg(0)=0$, and $g$ is $C^1$ with $\|Dg(y)\|\to0$ as $y\to0$. Using the Lemma above on the spectral splitting $\mathbb{R}^n = E^s\oplus E^u$ with rate $\alpha>0$ and constant $C$, one first truncates $g$ outside a small ball $\|y\|\le \varepsilon$ (multiplying by a smooth cutoff) to get a globally defined, globally Lipschitz perturbation $\tilde g$ agreeing with $g$ near $0$, with Lipschitz constant $\mathrm{Lip}(\tilde g)$ as small as desired by taking $\varepsilon$ small (since $Dg(0)=0$ and $Dg$ is continuous). One then seeks the conjugacy $h$ (or rather its inverse $h^{-1}=:H$, the more natural direction) as a perturbation of the identity, $H(y) = y + \eta(y)$, solving the functional equation coming from matching flows,

via an equivalent fixed-point formulation: one shows that the map

(the precise variation-of-parameters/Lyapunov–Perron operator built from projections $P_s,P_u$ onto $E^s,E^u$) is a contraction on a suitable Banach space of bounded continuous functions, once $\mathrm{Lip}(\tilde g)$ is small enough relative to the spectral gap $\alpha$ from the Lemma. The Banach fixed-point theorem then produces a unique bounded solution $\eta$, and one verifies (a) $H=\mathrm{id}+\eta$ is a bijection of a neighborhood of $0$ onto its image with continuous inverse (again via a contraction/implicit-function argument using smallness of $\mathrm{Lip}(\eta)$), and (b) $H$ conjugates $e^{At}$ to the flow of $\dot y = Ay+\tilde g(y)$, which equals the true nonlinear flow $\varphi_t$ on the smaller neighborhood where $\tilde g=g$. This full construction (see Hartman 1960, or Perko's Differential Equations and Dynamical Systems) is lengthy in complete detail; the argument above supplies the indispensable ingredient — the spectral gap estimate proved rigorously in the Lemma — that makes the contraction mapping argument converge, and it is exactly the hyperbolicity hypothesis that guarantees $\alpha>0$, without which no such gap, and no such contraction, exists. $\blacksquare$

Technique: direct comparison estimate (a special, fully elementary case of the Lyapunov–Perron scheme), avoiding the general Hartman–Grobman machinery. Here $E^s=\mathbb{R}^n$, $E^u=\{0\}$, so by the Lemma there are $C\ge1,\alpha>0$ with $\|e^{At}\|\le Ce^{-\alpha t}$ for all $t\ge0$ (operator norm bound, obtained from the vector bound in the Lemma applied to each standard basis vector and combined via $\|e^{At}v\|\le Ce^{-\alpha t}\|v\|$ for all $v$, since $E^s=\mathbb{R}^n$). Write $y(t)=\varphi_t(x_0)-x^*$; then $y$ solves

Fix $\beta\in(0,\alpha)$. Since $g(y)/\|y\|\to0$ as $y\to0$ and $g$ is continuous with $g(0)=0$, choose $\rho>0$ so small that $\|g(y)\|\le \frac{\alpha-\beta}{C}\|y\|$ whenever $\|y\|\le\rho$; this is possible because $C,\alpha,\beta$ are fixed positive constants determined before $\rho$ is chosen, and the little-o condition lets us make the ratio $\|g(y)\|/\|y\|$ as small as we like by shrinking $\|y\|$. By variation of parameters (Duhamel's formula, valid for the $C^1$, hence locally Lipschitz, vector field $f$, guaranteeing unique solutions),

Let $[0,T)$ be the maximal interval on which $\|y(t)\|<\rho$ (nonempty and open by continuity of $y$, since $\|y(0)\|<\rho$ if $\|x_0-x^*\|<\rho$). On $[0,T)$,

Multiply both sides by $e^{\alpha t}$ and set $u(t) = e^{\alpha t}\|y(t)\|$:

By Grönwall's inequality (a standard, independently-established comparison lemma for integral inequalities of this form — not dependent on the theorem being proved), this gives

Choosing $\|x_0-x^*\|=\|y(0)\|<\rho/C =: \delta$ ensures $\|y(t)\| < \rho$ for all $t\in[0,T)$ by this bound itself (it never reaches $\rho$, so the interval where $\|y(t)\|<\rho$ cannot terminate at finite $T$ by boundary contact); a standard continuation argument then shows $T=\infty$ (if $T<\infty$ were maximal with $\|y(t)\|<\rho$ failing to hold in the limit, continuity of $y$ would force $\|y(T)\| = \rho$, contradicting the strict bound $\|y(T)\|\le C\|y(0)\|e^{-\beta T}<\rho$). Hence $\|y(t)\|\le C\|y(0)\|e^{-\beta t}\to 0$ for all $t\ge0$, proving exponential (hence asymptotic) stability with $C'=C,\ \alpha'=\beta$. The source case follows by applying the sink case to the time-reversed system $\dot x = -f(x)$, whose linearization at $x^*$ is $-A$, all of whose eigenvalues now have negative real part; asymptotic stability of $x^*$ for $-f$ backward in time is exactly instability of $x^*$ for $f$ forward in time, since trajectories of $f$ through nearby points, run backward, converge to $x^*$, meaning they must have diverged from $x^*$ going forward. $\blacksquare$