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Floquet's theorem states that if $A(t)$ is a continuous matrix function with period $T$ then the solution $\Phi(t)$ satisfying the equation $\partial_t x(t) = A(t) x(t)$ can be written in the following decomposition $$ \Phi(t) = P(t) e^{Bt} $$ where $P(t)$ is a nonsingular, continguously differentiable matrix function with period $T$ and $B$ a constant, possibly complex matrix.

Proof can be found here for example https://proofwiki.org/wiki/Floquet%27s_Theorem

My question is, is there a higher dimension analog of Floquet's theorem? namely consider $\vec{t} = (t_1, t_2 \cdots, t_n)$ and $\sum_i \partial_{t_i}x(\vec{t}) = A(\vec{t}) x(\vec{t})$, where $A(\vec{t})$ is periodic in each argument $t_i$ with period $T_i$ respectively. Assume the periods are incommensurate.

Does an analogous decomposition hold for the solution $\Phi(t)$, and/or what properties does it have? If not, a partial answer as to what the structure of $\Phi(t)$ could be, would also be useful. Naively I don't see how the proofs in the 1d case straightforwardly generalize. Thanks!

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  • $\begingroup$ It seems like this would depend on whether the LCM of the $t_i$ is finite or not. $\endgroup$
    – Ian
    Commented Jul 6, 2018 at 20:39
  • $\begingroup$ indeed, if the LCM is finite then we're back in the realm of a periodic function with single period, and the standard Floquet theorem applies. The more interesting case is when the periods/frequencies are incommensurate. $\endgroup$
    – nervxxx
    Commented Jul 6, 2018 at 20:47
  • $\begingroup$ Try Googling quasi-periodic systems, reducibility, Lyapunov-Perron transformation. $\endgroup$
    – user539887
    Commented Jul 7, 2018 at 11:16
  • $\begingroup$ I see, essentially the answer is that it's still an open, important question aside from minor partial results $\endgroup$
    – nervxxx
    Commented Jul 7, 2018 at 21:56

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