On the Dynamics Around the Collinear Points in the Sun-Jupiter System

[eng] This work aims the study of the Rapid Transition Mechanism that explains some properties of orbits of some spatial objects, as for instance, comet 39P/Oterma, which will be the main object of this research. Considering Sun and Jupiter are the masses that more influence the considered object, this mechanism describes a transition which makes the object to change from an orbit which is outside the Jupiter's one from one inside of it or viceversa. This mechanism is observed, in particular, in the phase space of the considered models: the Restricted Three-Body Problem, both Planar Circular and Planar Elliptic. In these models three bodies are considered, two of them, named primaries, have positive mass and their orbits evolve according to the solution of the Two-Body Problem, i.e., they are circles, ellipses, parabolas or hyperbolas, having as focus (or centre) the centre of mass of both masses. The third body (which movement is to be described) is considered to have zero mass, hence it does not influence the movement of the primaries but it is under their gravitational influence. We will present the cases on which the orbit of the primaries is a circle or an ellipse and that the orbit of the third body is confined to the same plane of movement of the primaries. Having chosen the models to study this type of transition, we proceed to the study of the skeletons of these systems, i.e., which invariant objects are the more important and responsible to describe Oterma's dynamics. This methodology is general in the study of the phase space of dynamical system: these objects are equilibrium points, periodic orbits, tori, manifolds, atractors, repulsors, among others, based on each problem. To compute the equilibrium points L1 and L2 in the circular model (which will be also used in the elliptic one) it is enough to numerically solve a polynomial equation of 5th degree, known as Euler's quintic. Afterwards, the periodic orbits around them are computed via two approaches: a semi-analytical one (which also permit the compute a good initial approximation of their stable and unstable invariant manifolds) using Birkhoff Normal Forms at the equilibirum points and a numerical one. In the elliptic model, the tori around L1 and L2 are computed using numerical techniques, approximating a parameterization using Fourier series. In fact, it is considered the mapping as integrating a period of Jupiter and an invariant curve can be computed. Due to the strong instability of the region around the equilibirum points, we consider a representation using more than 1 section in the independent variable and the 1-period-integration is done in smaller steps - this approach is called parallel shooting. Finally, we visualize Oterma in this context. Changes of variable are done in order to fit its real data in both model. This lets us read Oterma's positions and velocities from JPL-Horizons system and represent them in synodical coordinates. Approximating the initial coordinates (projecting them in the primaries plane of movement) and integrating them in the planar elliptic model we obtain a good hint that this model is suitable to reproduce, at least partially, Oterma's dynamics. With this, we are able to visualize Oterma inside the phase space and how it interacts with the considered invariant objects. In particular, making sections in the true anomaly and in the x coordinate at the same time, it is possible to compute invariant tori around L1 and around L2 which invariant manifolds are closer to Oterma's orbit. In addition, still in these double sections we are able to visualize the heteroclinic connections between these tori near Oterma's orbit

Using normal forms to study Oterma's transition in the Planar RTBP

Comet 39P/Oterma is known to make fast transitions between heliocentric orbits outside and inside the orbit of Jupiter. In this note the dynamics of Oterma is quantitatively studied via an explicit computation of high order Birkhoff normal forms at the points $L_1$ and $L_2$ of the Planar Restricted Three-Body Problem. A previous work [14] has shown the existence of heteroclinic connections between the neigbourhood of $L_1$ and $L_2$ which provide paths for this transition. Here we combine real data on the motion of Oterma with normal forms to compute the invariant objects that are responsible for this transition

N-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics

International audienceThe formulation of the dynamics of N-bodies on the surface of an infinite cylinder is considered. We have chosen such a surface to be able to study the impact of the surface's topology in the particle's dynamics. For this purpose we need to make a choice of how to generalize the notion of gravitational potential on a general manifold. Following Boatto, Dritschel and Schaefer [5], we define a gravitational potential as an attractive central force which obeys Maxwell's like formulas. As a result of our theoretical differential Galois theory and numerical study - Poincare sections, we prove that the two-body dynamics is not integrable. Moreover, for very low energies, when the bodies are restricted to a small region, the topological signature of the cylinder is still present in the dynamics. A perturbative expansion is derived for the force between the two bodies. Such a force can be viewed as the planar limit plus the topological perturbation. Finally, a polygonal configuration of identical masses (identical charges or identical vortices) is proved to be an unstable relative equilibrium for all N > 2

Multi-objective approach for multiple clusters detection in data points events.

The spatial scan statistic is a widely used technique for detecting spatial clusters. Several extensions of this technique have been developed over the years. The objectives of these techniques are the detection accuracy improvement and a flexibilization on the search clusters space. Based on Voronoi-Based Scan (VBScan), we propose a biobjective approach using a recursively VBScan method called multiobjective multiple clusters VBScan (MOMC-VBScan), alongside a new measure called matching. This approach aims to identify and delineate all multiple significant anomalies in a search space. We conduct several experiments on different simulated maps and two real datasets, showing promising results. The proposed approach proved to be fast and with good precision in determining the partitions

Spatial cluster analysis using particle swarm optimization and dispersion function.

Spatial patterns studies are of great interest to the scientific community and the spatial scan statistic is a widely used technique to analyze such patterns. A key point for the construction of methods for detection of irregularly shaped clusters is that, as the geometrical shape has more degrees of freedom, some correction should be employed in order to compensate the increased flexibility. This paper proposed a multi-objective approach to cluster detection problem using the Particle Swarm Optimization technique aggregating a novel penalty function, called dispersion function, allowing only clusters which are subsets of a circular zone of moderate size. Compared to other regularity functions, the multi-objective scan with the dispersion function is faster and suited for the detection of moderately irregularly shaped clusters. An application is presented using statewide data for Chagas? disease in puerperal women in Minas Gerais state, Brazil