arXiv subjects

Cengiz Aydin

Publications and source records attributed to Cengiz Aydin.

6 recordsLinked to original sources

arXiv2026

Comet-type periodic motions and their out-of-plane bifurcations in the Earth-Moon CR3BP: a computational symplectic analysis

Comet-type periodic orbits of the circular restricted three-body problem (CR3BP) are periodic solutions that are generated from very large retrograde and direct circular Keplerian motions around the common center of mass of the primaries. In this paper we first provide an analytical proof of the existence of the comet-type periodic orbits by using the classical Poincaré continuation method. Within this analytical approach, we also determine the Conley-Zehnder index, defined as a Maslov index using a crossing form. Then, by applying a standard corrector-predictor technique, we explore numerically the two families of comet orbits within the Earth-Moon CR3BP. We compute their stability indices, identify vertical self-resonant bifurcations of higher order periods (of multiplicity from integer multiples up to six), investigate the vertically bifurcated spatial periodic solutions, and discuss their orbital characteristics. We also describe the orbits that are in resonance with the Earth and the Moon. We illustrate our main results in the form of bifurcation graphs, based on symplectic invariants, that provide a topological overview of the connections of the bifurcated branches, including bridge families.

math.SG↗

arXiv2026

Exploration of vertical self-resonant bifurcations from Distant Retrograde Orbits (DROs) in the Earth-Moon Circular Restricted Three-Body Problem (CR3BP)

The purpose of this paper is to investigate vertical self-resonant (VSR) bifurcations from the distant retrograde orbit (DRO) family in the framework of the Earth-Moon circular restricted three-body problem (CR3BP). To this end, by using a classical corrector-predictor algorithm we compute the vertical stability of the DROs and identify fourteen vertical-critical DROs. We split them into three groups according to orbiting around the libration points $L_i$, $i=1,2,4,5$. (i) We first analyze six VSR bifurcations of higher order periods (of multiplicity from integer multiples of five to ten) associated with the DROs near the Moon. (ii) For the DROs that move near the Moon and additionally around the $L_1$ and $L_2$ libration points, we study six VSR bifurcations of multiplicity from five to ten as well. (iii) Within the DROs orbiting around the $L_4$ and $L_5$ libration points, two vertical single-turn branch points occur. In total, we generate 25 bifurcated families of spatial symmetric periodic solutions and present their orbital characteristics, including bridge families to the Butterfly, prograde orbits, quasi DROs and DROs. We also obtain branches whose members consist of long periods combining almost planar ecliptic motions with several spatial excursions, during which the trajectory repeatedly moves far from and then close to the Moon, being one of Bumble Bee, Hoverfly or Dragonfly shape. We also find spatial orbits that are in resonance with the Earth and the Moon. In order to provide a structured and systematic overview of such bifurcation results, we determine Conley-Zehnder indices and construct bifurcation diagrams in view of symplectic invariants.

math.DS↗

arXiv2025

Contact geometry of Hill's approximation in a spatial restricted four-body problem

It is well-known that the planar and spatial circular restricted three-body problem (CR3BP) is of contact type for all energy values below the first critical value. Burgos-García and Gidea extended Hill's approach in the CR3BP to the spatial equilateral CR4BP, which can be used to approximate the dynamics of a small body near a Trojan asteroid of a Sun--planet system. Our main result in this paper is that this Hill four-body system also has the contact property. In other words, we can "contact" the Trojan. Such a result enables to use holomorphic curve techniques and Floer theoretical tools in this dynamical system in the energy range where the contact property holds.

math.SG↗

arXiv2025

Studying network of symmetric periodic orbit families of the Hill problem via symplectic invariants

In the framework of the spatial circular Hill three-body problem we illustrate the application of symplectic invariants to analyze the network structure of symmetric periodic orbit families. The extensive collection of families within this problem constitutes a complex network, fundamentally comprising the so-called basic families of periodic solutions, including the orbits of the satellite $g$, $f$, the libration (Lyapunov) $a,c$, and collision $\mathcal B_0$ families. Since the Conley-Zehnder index leads to a grading on the local Floer homology and its Euler characteristics, a bifurcation invariant, the computation of those indices facilitates the construction of well-organized bifurcation graphs depicting the interconnectedness among families of periodic solutions. The critical importance of the symmetries of periodic solutions in comprehending the interaction among these families is demonstrated.

math.DS↗

arXiv2024

Symplectic geometry and space mission design

Using methods from symplectic geometry, the second and fifth authors have provided theoretical groundwork and tools aimed at analyzing periodic orbits, their stability and their bifurcations in families, for the purpose of space mission design. The Broucke stability diagram was refined, and the "Floer numerical invariants" where considered, as numbers which stay invariant before and after a bifurcation, and therefore serve as tests for the algorithms used. These tools were later employed for numerical studies. In this article, we will further illustrate these methods with numerical studies of families of orbits for the Jupiter-Europa and Saturn-Enceladus systems, with emphasis on planar-to-spatial bifurcations, from deformation of the families in Hill's lunar problem studied by the first author. We will also provide an algorithm for the numerical computation of Conley--Zehnder indices, which are instrumental in practice for determining which families of orbits connect to which. As an application, we use our tools to study a family of periodic orbits that approaches Enceladus at an altitude of 29km, and therefore may be used in future space missions to visit the water plumes.

math.SG↗

arXiv2022

From Babylonian lunar observations to Floquet multipliers and Conley-Zehnder Indices

The lunar periods of our moon -- the companion of the Earth -- which date back to the Babylonians until around 600 BCE, are 29.53 days for the synodic, 27.55 days for the anomalistic and 27.21 days for the draconitic month. In this paper we define and compute these periods in terms of Floquet multipliers and Conley--Zehnder indices for planar periodic orbits in the spatial Hill lunar problem, which is a limit case of the spatial circular restricted three body problem. For very low energies, we are able to prove analytically the existence of the families of planar direct (family $g$) and retrograde periodic orbits (family $f$) and to determine their Conley-Zehnder index. For higher energies, by numerical approximations to the linearized flow, we also study other families of planar and spatial periodic orbits bifurcating from the families $g$ and $f$. Moreover, our framework provide an organized structure for the families, especially to see how they are connected to each other. Since the solutions we analyze are of practical interest, our work connects three topics: Babylonian lunar periods, symplectic geometry, and space mission design.

math.DS↗