(For convenience, below we assume the stationary point is the original point $O$.)
According to my understanding (not sure if it is correct), Poincaré linearization theorem says that, if the real parts of eigenvalues of $Df(0)$ are all positive/negative, then a nonlinear ode
$$\dot x = f(x) = A x + o(|x|^2) \quad\mathrm(Equ1)$$
can be transformed, by nonlinear coordinate transformation (i.e. $y = x + o(|x|^2)$), to a linear ode
$$\dot y = B y \quad\mathrm(Equ2);$$
in other words, the (un)stable manifolds of Equ1 can be mapped to (un)stable manifold of Equ2, with only a change of coordinate system/chart.
For example, we have a nonlinear ode about $x\in \mathbb{R}^2$, and its $Df(0)$ has two positive (or both negative) eigenvalues, then we would have a phase portrait (showing an (un)stable manifold $\mathscr{M_1}$) looking like a 'distorted' star or a swirl, and the phase portrait (showing an (un)stable manifold $\mathscr{M_2}$) of the corresponding linear ode would look similar but less distorted. And $\mathscr{M_1}$ and $\mathscr{M_2}$ can be directly linked by finding a homeomorphism between the two manifolds (i.e. transformation of coordination system)
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As for the (un)stable manifold theorem, it says that, if the real parts of eigenvalues of $Df(0)$ are all nonzero, then Equ1 would, locally (not necessarily in the neighborhood of the stationary point?), have (un)stable manifold $\mathscr{M_3}$ of the same dimensions as those $\mathscr{M_4}$ of Equ2, and $\mathscr{M_3}$ would be tangent to $\mathscr{M_4}$.
For example, we have a nonlinear ode about $x\in \mathbb{R}^2$, and its $Df(0)$ has one positive and one negative eigenvalue, and the phase portrait would show that $\mathscr{M_3}$ and $\mathscr{M_4}$ are both saddles. $\mathscr{M_3}$ (stable/unstable manifold near the axis $x_1$/$x_2$), would be tangent to the axis $x_1$/$x_2$, which is exactly $\mathscr{M_4}$. (For a diagram, see https://physics.stackexchange.com/questions/465489/intuition-behind-manifold)
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Questions:
So can we say that the (un)stable manifold theorem extends the Poincaré linearization theorem from $Df(0)$ with all positive/negative eigenvalues to that with just nonzero eigenvalues, and from star- and swirl- shape (un)stable manifolds to (un)stable manifolds including those of saddle shapes?
Can $\mathscr{M_3}$ and $\mathscr{M_4}$ (in the saddle example) be directly linked by a homeomorphism between the two manifolds/a transformation of charts (in the sense of diff geom)? (According to Hartman-Grobman theorem--$Re(\lambda_i)\neq 0$ guarantees the homeomorphism of the neighborhoods of the stationary points of nonlinear ode and of its linearization--mentioned in Strogatz Nonlinear Dyanmics and Chaos 6.3, the answer seems yes.)
Poincaré linearization theorem also says that a nonlinear ode, the eigenvalues of whose $Df(0)$ satisfying a Siegel condition, can be transformed to a linear ode by just (nonlinear) coordinate transformation. How to understand the Siegel condition intuitively?
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References:
Glendinning, Stability, Instability and Chaos.