It is said in P. Holmes's book,
but how Poincaré maps is related to Floquet theory?
In my view, both theories are about decoupling a flow $\phi(x,t)$ into the periodic or fluctuating component $e^{i\theta(t)}$ and the amplitude component $A(t)$, and focusing on the amplitude component.
Poincaré maps do so by using a surface of section (manifold), so we observe only one or a few points from t to t+T (see https://physics.stackexchange.com/a/141539/273056); Floquet theory does so by using Floquet multiplier to separate the solution $x(t)$ to $v(t)e^{\sigma t}x_0$ ( e.g. let $v(t) = \phi(x,t) e^{-\sigma t}$, then $v(t+T) = v(t)$ using $\phi(x,t+T) e^{-\sigma T} = \phi(x,t)$ ), where v(t) is the fluctuating component. Is such understanding correct?
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But there seems to be deeper relations between the two, as explained in the same textbook (which I do not quite understand):
So can we understand Floquet theory intuitively by using Poincaré maps? If so, how? For example, can we think that, with proper choice of a surface $\mathscr{M}$ of section, the distance on $\mathscr{M}$ between a point and the stationary point corresponds exactly to $e^{\sigma t}x_0$?
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The excerpts come from:
P. Holmes, Nonlinear Oscillattions, Dynamical Systems and Bifurcations of Vector Fields, 1.5