Here $d$ is the degree of freedom. It is not the case when $d \geq 3$? Can anyone give an intuitive explanation?
When $d =2$, the phase space is 4 dimensional. The tori are 2 dimensional. In this case, a torus necessarily divide the phase space into two disjoint parts? It is the case when the ambient space is 3 dimensional, obviously.
They don't divide the phase space into disjoint parts. Rather, they divide the level set of a Hamiltonian system into disjoint parts, prevent the possibility of Arnold diffusion (trajectories drifting arbitrarily far and experiencing a significant change in action variable).
The phase space of a 2 d.o.f. Hamiltonian $H$ is four-dimensional; the trajectories are confined to exploring the level set $M$ given by $H=c$. The level set is invariant and three-dimensional. Hence, the two-dimensional KAM tori indeed divide the three dimensional level set and the trajectories are confined between two such level sets. For $d>2$, the dimension of $M$ is $2d-1 \geq 5$ and the $d$-dimensional KAM tori are no longer an obstacle to diffusion.
