Dr. Vidarte-Olivera, Jhon

We investigate the qualitative characteristics of a test particle attracted to an irregular elongated body, modeled as a non-homogeneous straight segment with a variable linear density. By deriving the potential function in closed form, we formulate the Hamiltonian equations of motion for this system. Our analysis reveals a family of periodic circular orbits parameterized by angular momentum. Additionally, we utilize the axial symmetry resulting from rotations around the segment’s axis to consider the corresponding reduced system. This approach identifies several reduced-periodic orbits by analyzing appropriate Poincaré sections. These periodic orbits are then reconstructed into quasi-periodic orbits within the full dynamical system.

This work investigates the qualitative dynamics of a massless particle around elongated bodies modeled with a linearly varying density function. We assume that the segment rotates uniformly with angular velocity w, with the fixed segment considered as a special case. For the rotating straight segment, we identify four equilibria: two collinear and two triangular. The primary novelty of our work is the detection of a pitchfork bifurcation, wherein, under slow and moderate rotation, the triangular equilibria shift toward and merge with one of the collinear points, while remaining distinct in fast rotations. This feature, previously unreported, is visualized through Hill’s regions and highlights how both rotation rate and asymmetry in mass distribution affect orbital dynamics, providing new insights into gravitational environments near irregularly shaped bodies. Additionally, we analyze the linear stability of the relative equilibria. Finally, for the fixed straight segment, we examine singular solutions, showing that any solution not defined globally over time leads the particle to eventually approach the segment at a distance of zero. In the one-dimensional problem, all singularities are observed to result from collisions.

We analyse the dynamics of an infinitesimal particle around an elongated body, which is modelled as a homogeneous fixed straight segment centred at the origin. We assume that the length of the segment is small compared with the distance to the particle. After a Lie–Deprit normalization, we end up with a Hamiltonian that has not only the mean anomaly but also the argument of the perigee relegated to terms or third order or higher. We employ invariant and reduction theories to reduce the artificial symmetries associated with the Kepler flow and the central action of the angular momentum. Analysing the relative equilibria in the first and second reduced spaces allows us to determine the existence of near-polar circular periodic orbits and KAM tori.

We construct a symplectic atlas adapted to the flow action of an uncoupled isotropic n-oscillator, referred to as the Reeb atlas. In the context of Reeb's Theorem for Hamiltonian systems with symmetry, these variables are very useful for finding periodic orbits and determining their stability in perturbed harmonic oscillators. These variables separate orbits, meaning they are in bijective correspondence with the set of orbits. Hence, they are especially suited for determining the exact number of periodic solutions via reduction and averaging methods. Moreover, for an arbitrary polynomial perturbation, we provide lower and upper bounds for the number of periodic orbits according to the degree of the perturbation.

We propose a novel numerical test to evaluate the reliability of numerical propagations, leveraging the fiber bundle structure of phase space typically induced by Lie symmetries, though not exclusively. This geometric test simultaneously verifies two properties: (i) preservation of conservation principles, and (ii) faithfulness to the symmetry-induced fiber bundle structure. To generalize the approach to systems lacking inherent symmetries, we construct an associated collective system endowed with an artificial G-symmetry. The original system then emerges as the G-reduced version of this collective system. By integrating the collective system and monitoring G-fiber bundle conservation, our test quantifies numerical precision loss and detects geometric structure violations more effectively than classical integral-based checks. Numerical experiments demonstrate the superior performance of this method, particularly in long-term simulations of rigid body dynamics and perturbed Keplerian systems.

We investigate the existence of periodic orbits in a perturbed Hamiltonian system of stellar type in 1:1 resonance. The perturbation consists of a potential of degree four with two real parameters. We determine six families of periodic orbits using reduction and averaging theories. Also, we characterize the stability of these orbits and their bifurcation curves in terms of the parameters. Finally, we show a complete picture of the choreographies of critical points originating the periodic orbits.

We study a type of perturbed polynomial Hamiltonian system in 1:1 resonance. The perturbation consists of a homogeneous quartic potential invariant by rotations of 𝜋/2 radians. The existence of periodic solutions is established using reduction and averaging theories. The different types of periodic solutions, linear stability, and bifurcation curves are characterized in terms of the parameters. Finally, some choreography of bifurcations are obtained, showing in detail the evolution of the phase flow.

We investigate analytically the existence of several families of periodic solutions for the planar anisotropic Schwarzschild-type problem. We use reduction and averaging theory, as well as the technique of continuation of Poincaré, for the study of symmetric periodic solutions. Moreover, the determination of KAM 2-tori encasing some of the linearly stable periodic solutions is proved. Finally, we analyze the existence of periodic Hamiltonian pitchfork bifurcation of the periodic solutions.

This article deals with the autonomous two-degree-of-freedom Hamiltonian system with Armbruster –Guckenheimer –Kim galactic potential in 1:1 resonance depending on two parameters. We detect periodic solutions and KAM 2-tori arising from linearly stable periodic solutions not found in earlier papers. These are established by using reduction, normalization, averaging and KAM techniques.