Poincare found these exponentially small effects in [Po]. In his study of periodic
orbits in two degrees of freedom Hamiltonian systems he proposed a model, which
after reduction became the following perturbed pendulum
y =
sin y + 2\ie cos y cos t
(using the same notation as Poincare). He deduced that the splitting of separatri-
ces is exponential small in /i, provided that e is less than an exponentially small
Arnold [Arl] found the exponentially small splitting of separatrices associated
to partially hyperbolic tori, studying the diffusion of action variables in near inte-
grable systems ho(I) + eh\ (£,/, e).
Neishtadt [Ne] gave upper bounds for the splitting in one and a half and two
degrees of freedom Hamiltonian systems with only one parameter.
Differential equations with slow dynamics are related with near the identity
diffeomorphisms by means of the Poincare map. It turns out that being Hamiltonian
is very important to get exponentially smallness. The Hamiltonian character of the
equation is translated to the symplectic character of the maps.
In [Lai] Lazutkin studied the standard map F(x, y) = (x+y+e sin x, y+e sinx)
and provided the following formula for the angle between the stable and unstable
manifolds at a homoclinic point
(2) p=^\Q1\e-*a/^[l + 0(eb)]
with 0 b 1/8. This was the first exponentially small asymptotic formula for a
nontrivial problem with only one parameter. Although the proof was not complete
Lazutkin introduced pioneering new analytic tools for the study of the separatrix
splitting which have decisively influenced the development of this area.
Several papers deal with the computation of the constant 0 i [LST] [Su]. The
complete proof of (2) is in [Ge4].
Fontich and Simo [FS1] [FS2] study the splitting of separatrices for families of
diffeomorphisms in a neighborhood of the identity of class C^ and Cr respectively.
Under fairly general hypotheses exponentially small upper bounds are obtained for
the distance between invariant manifolds in the analytic case with generally optimal
values of the constant in the exponent.
Other works referring to maps are [Ch] [DR2] [DR1] [Ge5] [GS].
Many authors have studied the phenomenon of separatrix splitting with fast
frequency periodic perturbations, in order to prove that in certain cases the Mel-
nikov function yields the right asymptotics of the measure of separatrix splitting.
They consider
i - f(z) + nepg{x, t/e, //), zeU CM2,
where /i and e 0 are parameters a priori independent and such that the origin is a
saddle-type fixed point. There has been a lot of discussion about the optimal value
of p for which one gets exponentially small upper bounds or asymptotics. In [Fo2]
upper bounds for the splitting are given even for negative values of p, specifically
p —1/2. If the model is simplified, considering equations of the form
(3) x + f(x) = fiepg(x,t/s,s,fi), x e V c M ,
then in [Fol] upper bounds are given for the splitting of separatrices for values of
p —2. To ask the perturbation to have order /iep, with p bigger than some value,
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