  This work studies existence and regularity questions for attracting invariant
tori in three dimensional dissipative systems of ordinary differential
equations. Our main result is a constructive method of computer assisted proof
which applies to explicit problems in non-perturbative regimes. We obtain
verifiable lower bounds on the regularity of the attractor in terms of the
ratio of the expansion rate on the torus with the contraction rate near the
torus. We consider separately two important cases of rotational and resonant
tori. In the rotational case we obtain $C^k$ lower bounds on the regularity of
the embedding. In the resonant case we verify the existence of tori which are
only $C^0$ and neither star-shaped nor Lipschitz
