One of the main tools for determining the existence of (or non-existence of) chaos in a perturbed Hamiltonian system is Melnikov theory. In this theory, the distance between the stable and unstable manifolds of the perturbed system is calculated up to the first-order term. Consider a smooth dynamical system
x
¨
=
f
(
x
)
+
ϵ
g
(
t
)
, with
ϵ
≥
0
and
g
(
t
)
periodic with period
T
. Suppose for
ϵ
=
0
the system has a hyperbolic fixed point x0 and a homoclinic orbit
ϕ
(
t
)
corresponding to this fixed point. Then for sufficiently small
ϵ
≠
0
there exists a T-periodic hyperbolic solution. The stable and unstable manifolds of this periodic solution intersect transversally. The distance between these manifolds measured along a direction that is perpendicular to the unperturbed homoclinc orbit
ϕ
(
t
)
is called the Melnikov distance. If
d
(
t
)
denotes this distance, then
d
(
t
)
=
ϵ
(
M
(
t
)
+
O
(
ϵ
)
)
. The function
M
(
t
)
is called the Melnikov function.
