x I. BALDOMA AND E. FONTICH
The memoir is organized as follows. In the first chapter we introduce the
notation, the hypotheses and the main theorem.
In the second chapter, we study some analytical properties of the homoclinic
orbit of the unperturbed system. In particular we get the asymptotic behavior of its
parameterization. We prove that, as was to be expected, this behavior is algebraic,
that is, there exists T 0 such that if t G C, Ret T, the stable manifold behaves
like l/tp with p a certain positive number, and, analogously, the unstable manifold
has the form l/(-t)p for t G C, Ret - T .
In the third chapter, we establish, under the stated conditions, the existence
of stable and unstable invariant manifolds for the perturbed system. Moreover we
find useful parameterizations 7*(t, s) (* = s,u), of the local invariant manifolds of
the perturbed Hamiltonian system. These parameterizations satisfy that 7* is a
solution with respect to the variable t G R, it is analytic with respect to s and
7*(t + 27T£,s) - 7 * ( £ , S + 27T£).
In this way we endow the variable s with a dynamic character since, if Pto is the
Poincare map from t = to to t to +
2TT£,
7s(to, s) represents the stable manifold
of Pto and the dynamics of Pto on it is simply
Pto(7s(t0,s))
=
ys(t0,s
+ 2ne).
Moreover 7* is of the form
7*(t,S)=7o(t +
5
) + ^
+
V * ( t ,
S
) ,
where 70 is the homoclinic orbit of the unperturbed system.
In the fourth chapter, we built the flow box coordinates, i.e., coordinates in
which the flow straightens. These coordinates are defined in a neighborhood of the
stable manifold not containing the origin, but close to it and independent of the
parameters. We built them following several steps. We parameterize the solutions
of the perturbed system near a piece of the stable manifolds by two parameters.
One of these is time, and the other is a complex parameter s such that the solutions
are analytic with respect to s and the dynamics of the Poincare mapping is simply
s ^ s +
2TT£.
We can write them in the form w(t + s,t/s). We prove afterwards,
thanks to this good parameterization, that the solutions intersect a (real) section
transverse to the flow for some value (to, so), thus we are able to straighten the flow
in a neighborhood of the stable manifold. Finally we slightly modify the variables
to make them canonical.
In the fifth chapter, we present a result of Delshams-Seara [DS2] which asserts
that if p is bigger than some value, which depends on the perturbation and the
unperturbed homoclinic orbit, we can extend the parameterization of the unstable
manifold until it reaches the domain where the flow box variables are defined.
Finally, in the last chapter we introduce the splitting function. From it and its
properties we derive the asymptotic formulas for the area of the lobes generated by
the invariant manifolds between two homoclinic points and the angle between the
invariant manifolds at a homoclinic point. They are exponentially small in e. The
main difference with [DS2] is that here we consider homoclinic orbits with algebraic
branch type singularities and consequently some computations are somewhat more
involved.
Previous Page Next Page