# Splittings for generic flows on a vector bundle

Question: Consider a smooth vector bundle $$\pi:V\to B$$ and the space $$\mathcal{F}$$ of $$C^k$$ linear flows $$\Bbb{R}\times V \to V$$ endowed with the strong $$C^k$$ Whitney topology. Is it true that for all linear flows $$\Phi$$ in an open-dense (or perhaps residual) subset of $$\mathcal{F}$$, $$V$$ will split as a $$\Phi^t$$-invariant Whitney sum $$V = E^1 \oplus \cdots \oplus E^m$$

where each of the $$E^i$$ is a rank one or two vector subbundle? (And if true, is smoothness necessary -- e.g., can we set $$k = 0$$?)

Motivation: The constant matrix $$A$$ defining the linear ODE on $$\Bbb{R}^n$$:

$$\dot{x} = Ax$$

generically has distinct eigenvalues, and therefore $$\mathbb{R}^n$$ splits as a direct sum of one and two dimensional linear subspaces which are invariant under the induced flow.

Similarly, by Floquet theory it follows for a generic $$2\pi$$-periodic $$A(t)$$, the trivial bundle $$S^1\times \Bbb{R}^n$$ over $$S^1 = [0,2\pi]/\{0,2\pi\}$$ has a $$\Phi^t$$-invariant splitting into rank one or two subbundles, where $$\Phi:\Bbb{R}\times (S^1\times \Bbb{R}^n)$$ is the linear flow induced by

$$\dot{x} = A(t)x.$$