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2013
Impact Factor

R. Martinez

Barcelona, Spain
Dept. de Matematiques Universitat Autonoma de Barcelona, Bellaterra

Publications:

Martinez R., Simo C.
Invariant Manifolds at Infinity of the RTBP and the Boundaries of Bounded Motion
2014, vol. 19, no. 6, pp.  745-765
Abstract
Invariant manifolds of a periodic orbit at infinity in the planar circular RTBP are studied. To this end we consider the intersection of the manifolds with the passage through the barycentric pericenter. The intersections of the stable and unstable manifolds have a common even part, which can be seen as a displaced version of the two-body problem, and an odd part which gives rise to a splitting. The theoretical formulas obtained for a Jacobi constant $C$ large enough are compared to direct numerical computations showing improved agreement when $C$ increases. A return map to the pericenter passage is derived, and using an approximation by standard-like maps, one can make a prediction of the location of the boundaries of bounded motion. This result is compared to numerical estimates, again improving for increasing $C$. Several anomalous phenomena are described.
Keywords: invariant rotational curves, separatrix maps, splitting function, restricted three-body problem
Citation: Martinez R., Simo C.,  Invariant Manifolds at Infinity of the RTBP and the Boundaries of Bounded Motion, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 745-765
DOI:10.1134/S1560354714060112
Martinez R., Simo C.
Non-Integrability of Hamiltonian Systems Through High Order Variational Equations: Summary of Results and Examples
2009, vol. 14, no. 3, pp.  323-348
Abstract
This paper deals with non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to deal with these problems. We display a family of Hamiltonian systems which require the use of order $k$ variational equations, for arbitrary values of $k$, to prove non-integrability. Moreover, using third order variational equations we prove the non-integrability of a non-linear springpendulum problem for the values of the parameter that can not be decided using first order variational equations.
Keywords: non-integrability criteria, differential Galois theory, higher order variationals, springpendulum system
Citation: Martinez R., Simo C.,  Non-Integrability of Hamiltonian Systems Through High Order Variational Equations: Summary of Results and Examples, Regular and Chaotic Dynamics, 2009, vol. 14, no. 3, pp. 323-348
DOI:10.1134/S1560354709030010

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