In an informal way, Takens’ theorem states, that we can reconstruct a (chaotic) attractor of an $N$-dimensional dynamical system with a delay embedding of just one trajectory $x_i(t)$:
$(x_i(t),x_i(t-\tau),...,x_i(t-k \cdot \tau))$
for some $k \geq 2d$ has equivalent dynamics to
$(x_1(t),...,x_d(t))$.
Suppose we have a multistable dynamical system with e.g. two (chaotic) attractors. If we sample the trajectory on one attractor, can we recover the other one by delayed embedding?
Is there some known generalization of Takens’ theorem to multistable systems?
This is impossible.
If your dynamics is on one attractor, you can only learn about that attractor¹ from observing the dynamics. To see this, consider what would happen if you radically change the dynamics outside the immediate vicinity of the observed attractor, but keep the attractor identical: Your dynamics would not change the slightest, because your trajectory never ventures into the regions you changed anyway.
As a specific example consider a ball moving on some geography with two valleys. This is a textbook example of a bistable system. If I observe a ball oscillating in one valley, this tells me nothing about the geography outside that valley (or more precisely the regions the ball visits). I can completely remove the other valley, add further valleys, etc.: The ball would still oscillate in exactly the same manner.
¹ and its very close vicinity depending on the continuity assumptions you can make.