Kolmogorov theorem (KAM theory): If the unperturbed system is nondegenerate or isoenergetically nondegenerate, then for a sufficiently small Hamiltonian perturbation most nonresonant invariant tori do not vanish but are only slightly deformed, so that in the phase space of the perturbed system there are invariant tori densely filled with conditionally-periodic phase curves winding around them, with a number of independent frequencies equal to the number of degrees of freedom. These invariant tori form a majority in the sense that the measure of the complement of their union is small when the perturbation is small. In the case of isoenergetic nondegeneracy the invariant tori form a majority on each level manifold of the energy.
That visualization is of the restricted three-body problem for a particle (or asteroid) in the Sun-Jupiter system. Each frame is a Poincaré section at a different value of the Hamiltonian energy.
