Introduction
Consider a system with an invariant object (fixed point, periodic orbit, etc)
which has stable and unstable invariant manifolds associated to it and they co-
incide, or some branches of them coincide. If we perturb the system, generically
the invariant manifolds will not coincide any more. This phenomenon is known as
splitting of separatrices or splitting of invariant manifolds. One of the simplest set-
tings where this phenomenon occurs is in differential equations in the plane having
a hyperbolic saddle fixed point and a homoclinic connection associated to it. When
we perturb this system with a time periodic perturbation, say
z = f(z)+eg(z,t,e), z e U C
R2,
the fixed point becomes a hyperbolic periodic orbit with two dimensional stable
and unstable invariant manifolds in M3. Using a first order perturbation theory the
distance between the splitted manifolds measured in a plane {t = to} orthogonally
to the unperturbed homoclinic connection at some point p is given by
(1) d(t0,e) = ^j^e +
O(e2)
where M(to) is the so called Poincare-Melnikov function which is given through an
integral in terms of the system and the unperturbed homoclinic orbit. For systems
with slow dynamics such as
z = ef(z)+e2g(z,t,e)
we can scale time through et r and we obtain
z = f(z) + eg(z,r/£,s)
which is a perturbation of z = f(z). The formal substitution of g into the Poincare-
Melnikov function gives an ^-dependent function which is exponentially small in e
[Fo2]. Then, in (1) the
0(s2)
term dominates over £M(£o)/||/(p)|| and we do not
have an asymptotic expression of d(to,e). We only know that it is
0(e2).
One way to obtain rigorous asymptotic expressions is to introduce another
parameter, that is, to consider
z = f(z)+V9{z,t/e,p,).
Then
d(*0,/i,e) = ^ ^ M + O(M2)
and hence, if p is small enough compared with M(to,e), which is exponentially
small in £, we have that d ~ M(to,£)p/\\f(p)\\. But this only gives rigorous results
in a very narrow set in the space of parameters.
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