The 2-body problem under Fock's potential
Vasile Mioc Ernesto Pérez-Chavela
Discrete & Continuous Dynamical Systems - S 2008, 1(4): 611-629 doi: 10.3934/dcdss.2008.1.611
We study the two-body problem moving under the Fock's potential, where the global flow is fully described. The analysis is separately performed for negative, zero, and positive energy levels. Many kinds of orbits are found, some of them being of positive Lebesgue measure. We also show some unusual features as the coexistence of fundamentally different orbits for the same energy level and for the same angular momentum.
keywords: bifurcations. singularities homoclinic orbit regularization Fock's field
Zero-Hopf bifurcation for a class of Lorenz-type systems
Jaume Llibre Ernesto Pérez-Chavela
Discrete & Continuous Dynamical Systems - B 2014, 19(6): 1731-1736 doi: 10.3934/dcdsb.2014.19.1731
In this paper we apply the averaging theory to a class of three-dimensional autonomous quadratic polynomial differential systems of Lorenz-type, to show the existence of limit cycles bifurcating from a degenerate zero-Hopf equilibrium.
keywords: Three-dimensional differential systems Lorenz-type system. limit cycles zero-Hopf equilibrium
Horseshoe periodic orbits with one symmetry in the general planar three-body problem
Abimael Bengochea Manuel Falconi Ernesto Pérez-Chavela
Discrete & Continuous Dynamical Systems - A 2013, 33(3): 987-1008 doi: 10.3934/dcds.2013.33.987
Using collinear reversible configurations and some properties of symmetry we obtain horseshoe periodic orbits in the general planar three-body problem with masses $m_1\gg m_2 \geq m_3$, which usually represents a system formed by a planet and two small satellites; for instance, the system Saturn-Janus-Epimetheus. For the numerical analysis we have taken the values $m_2/m_1 = 3.5 \times 10^{-4}$ and $m_3/m_1 = 9.7 \times 10^{-5}$ corresponding to $10^5$ times the mass ratios of Saturn-Janus and Saturn-Epimetheus,
keywords: periodic orbits. General three-body problem horseshoe orbits
Amadeu Delshams Marian Gidea Ernesto Pérez-Chavela
Discrete & Continuous Dynamical Systems - A 2013, 33(3): i-i doi: 10.3934/dcds.2013.33.3i
The material of this special issue of DCDS-A was originally dedicated in honor of the 65-th birthday of Prof. Ernesto A. Lacomba. Some of the papers in this issue reflect the joyful spirit surrounding this celebration. Sadly, shortly after the preparation of this volume was completed, Prof. Ernesto A. Lacomba passed away on June 26, 2012. Therefore this special issue is also paying a tribute to his long standing mathematical legacy.
    The work of Prof. Lacomba comprised research on geometric theory of ordinary differential equations, dynamical systems, and symplectic geometry, with applications to celestial mechanics, classical mechanics, vortex theory, thermodynamics and electrical circuits. Prof.~Lacomba was the leader of a strong research group working in these areas. In 1991 he started organizing, jointly with some members of his group and with other collaborators, the International Symposium on Hamiltonian Systems and Celestial Mechanics (HAMSYS), which became a great success over the next several years. These symposia brought together top researches from several countries, working in the aforementioned topics, as well as many graduate students who had the opportunity to learn from and connect with the experts in the field, and often get inspiration and motivation to improve and finalize their doctoral theses.
    The framework for the celebration of the 65-th birthday of Prof. Lacomba was the VI-th edition of HAMSYS, which was held in México D.F. between November 29 -- December 3, 2010.
    This symposium assembled an impressive number of highly respected researches who generated important discussions among the participants, presented new problems, and identified future research directions. The emphasis of the talks was on Hamiltonian dynamics and its relationship to several aspects of mechanics, geometric mechanics, and dynamical systems in general. The papers in this volume are an outgrowth of the themes of the symposium. All papers that were submitted to this special issue underwent a through refereeing process typical to any top mathematical journal. The accepted papers form the present issue of DCDS-A.
    The symposium received generous support from CONACYT México and UAM-I. Special thanks are due to Universidad Autónoma Metropolitana for hosting the symposium in the beautiful colonial building Casa de la primera imprenta de América. Last but not least, we thank all participants for contributing to a week-long intense and highly productive mathematical experience.

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