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### Open Access Journals

DCDS-B

A classical problem in the study of the (conservative) unfoldings of the so called Hopf-zero bifurcation, is the computation of the splitting of a heteroclinic connection which exists in the symmetric normal form along the z-axis. In this paper we derive the inner system associated to this singular problem, which is independent on the unfolding parameter. We prove the existence of two solutions of this system related with the stable and unstable manifolds of the unfolding, and we give an asymptotic formula for their difference. We check that the results in this work agree with the ones obtained in the regular case by the authors.

DCDS

The restricted planar elliptic three body problem models the motion of a massless body under the Newtonian gravitational force of two other bodies, the primaries, which evolve in Keplerian ellipses.

A trajectory is called oscillatory if it leaves every bounded region but returns infinitely often to some fixed bounded region. We prove the existence of such type of trajectories for any values for the masses of the primaries provided the eccentricity of the Keplerian ellipses is small.

DCDS

We consider a singular or weakly hyperbolic Hamiltonian, with $n+1$ degrees of
freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near
a simple resonance. The model consists of an integrable Hamiltonian possessing
an $n$-dimensional hyperbolic invariant torus with fast frequencies
$\omega/\sqrt\varepsilon$ and coincident whiskers, plus a perturbation of order
$\mu=\varepsilon^p$. The vector $\omega$ is assumed to satisfy a Diophantine
condition.

We provide a tool to study, in this singular case, the splitting of the perturbed whiskers for $\varepsilon$ small enough, as well as their homoclinic intersections, using the Poincaré--Melnikov method. Due to the exponential smallness of the Melnikov function, the size of the error term has to be carefully controlled. So we introduce flow-box coordinates in order to take advantage of the quasiperiodicity properties of the splitting. As a direct application of this approach, we obtain quite general upper bounds for the splitting.

We provide a tool to study, in this singular case, the splitting of the perturbed whiskers for $\varepsilon$ small enough, as well as their homoclinic intersections, using the Poincaré--Melnikov method. Due to the exponential smallness of the Melnikov function, the size of the error term has to be carefully controlled. So we introduce flow-box coordinates in order to take advantage of the quasiperiodicity properties of the splitting. As a direct application of this approach, we obtain quite general upper bounds for the splitting.

DCDS

A sequence of "inner equations" attached to certain perturbations of the
McMillan map was considered in [5], their solutions were used
in that article to measure an exponentially small separatrix splitting.
We prove here all the results relative to these equations which are necessary
to complete the proof of the main result of [5].
The present work relies on ideas from resurgence theory: we describe the
formal solutions, study the analyticity of their Borel transforms and use
Écalle's alien derivations to measure the discrepancy between different
Borel-Laplace sums.

DCDS

The McMillan map is a one-parameter family of integrable symplectic
maps of the plane, for which the origin is a hyperbolic fixed point
with a homoclinic loop, with small Lyapunov exponent when the
parameter is small.
We consider a perturbation of the McMillan map for which we show that the loop
breaks in two invariant curves which are exponentially close one to the other
and which intersect transversely along two primary homoclinic orbits.
We compute the asymptotic expansion of several quantities related to
the splitting, namely the Lazutkin invariant and the area of the
lobe between two consecutive primary homoclinic points.
Complex matching techniques are in the core of this work.
The coefficients involved in the expansion have a resurgent origin, as shown
in [14].

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