DCDS
Oscillatory orbits in the restricted elliptic planar three body problem
Marcel Guardia Tere M. Seara Pau Martín Lara Sabbagh
Discrete & Continuous Dynamical Systems - A 2017, 37(1): 229-256 doi: 10.3934/dcds.2017009

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.

keywords: Restricted three body problem final motions oscillatory motions parabolic points lambda lemma
DCDS
The parameterization method for one- dimensional invariant manifolds of higher dimensional parabolic fixed points
Inmaculada Baldomá Ernest Fontich Rafael de la Llave Pau Martín
Discrete & Continuous Dynamical Systems - A 2007, 17(4): 835-865 doi: 10.3934/dcds.2007.17.835
We use the parameterization method to prove the existence and properties of one-dimensional submanifolds of the center manifold associated to the fixed point of $C^r$ maps with linear part equal to the identity. We also provide some numerical experiments to test the method in these cases.
keywords: Parabolic point parameterization method. invariant manifold
DCDS
Arnold diffusion in perturbations of analytic integrable Hamiltonian systems
Ernest Fontich Pau Martín
Discrete & Continuous Dynamical Systems - A 2001, 7(1): 61-84 doi: 10.3934/dcds.2001.7.61
Given an analytic integrable Hamiltonian with three or more degrees of freedom, we construct, arbitrarily close to it, an analytic perturbation with transition chains whose lengths only depend on the unperturbed Hamiltonian. Then we deduce that the perturbed system has Arnold diffusion. We provide the technical details of the tools we use.
keywords: Hamiltonian systems heteroclinic solutions invariant tori Arnold diffusion.
DCDS
Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points
Inmaculada Baldomá Ernest Fontich Pau Martín
Discrete & Continuous Dynamical Systems - A 2017, 37(8): 4159-4190 doi: 10.3934/dcds.2017177

We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to 1. We show that, under general hypotheses, these invariant manifolds are Gevrey with type related to some explicit constants. We provide examples of the optimality of our results as well as some applications to celestial mechanics, namely, the Sitnikov problem and the restricted planar three body problem.

keywords: Gevrey class parabolic points invariant manifolds
DCDS
Resurgence of inner solutions for perturbations of the McMillan map
Pau Martín David Sauzin Tere M. Seara
Discrete & Continuous Dynamical Systems - A 2011, 31(1): 165-207 doi: 10.3934/dcds.2011.31.165
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.
keywords: exponentially small phenomena splitting of separatrices. Resurgence
DCDS
Exponentially small splitting of separatrices in the perturbed McMillan map
Pau Martín David Sauzin Tere M. Seara
Discrete & Continuous Dynamical Systems - A 2011, 31(2): 301-372 doi: 10.3934/dcds.2011.31.301
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].
keywords: exponentially small phenomena splitting of separatrices asymptotic formula. McMillan map

Year of publication

Related Authors

Related Keywords

[Back to Top]