We study the splitting of invariant manifolds of whiskered t
ori with two or three frequencies in nearly-integrable
Hamiltonian systems. We consider 2-dimensional tori with a
frequency vector
ω
= (1
,
Ω) where Ω is a quadratic
irrational number, or 3-dimensional tori with a frequency v
ector
ω
= (1
,
Ω
,
Ω
2
) where Ω is a cubic irrational number.
Applying the Poincar ́e–Melnikov method, we find exponentia
lly small asymptotic estimates for the maximal splitting
distance between the stable and unstable manifolds associa
ted to the invariant torus, showing that such estimates
depend strongly on the arithmetic properties of the frequen
cies. In the quadratic case, we use the continued fractions
theory to establish a certain arithmetic property, fulfille
d in 24 cases, which allows us to provide asymptotic estimate
s
in a simple way. In the cubic case, we focus our attention to th
e case in which Ω is the so-called cubic golden number
(the real root of
x
3
+
x
−
1 = 0), obtaining also asymptotic estimates. We point out the
similitudes and differences
between the results obtained for both the quadratic and cubi
c cases.Preprin
