Computing Invariant Manifolds for Libration Point Missions

Mondelo, Josep–Maria

doi:10.1007/978-3-030-20633-8_4

Cited by 6 publications

(4 citation statements)

References 32 publications

(77 reference statements)

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“…System (22) has been solved by performing minimum-norm lest-squares Newton corrections, that can be computed through QR decompositions with column pivoting ( [19,29]) as long as the dimensions of the kernel is known. This methodology addresses the fact that number of equations and unknowns is different.…”

Section: Computing Heteroclinic Connectionsmentioning

confidence: 99%

Semi-analytical computation of heteroclinic connections between center manifolds with the parameterization method

Barcelona¹,

Haro²,

Mondelo³

2023

Preprint

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This paper presents methodology for the computation of whole sets of heteroclinic connections between iso-energetic slices of center manifolds of center × center × saddle fixed points of autonomous Hamiltonian systems. It involves: (a) computing Taylor expansions of the center-unstable and center-stable manifolds of the departing and arriving fixed points through the parameterization method, using a new style that uncouples the center part from the hyperbolic one, thus making the fibered structure of the manifolds explicit; (b) uniformly meshing iso-energetic slices of the center manifolds, using a novel strategy that avoids numerical integration of the reduced differential equations and makes an explicit 3D representation of these slices as deformed solid ellipsoids; (c) matching the center-stable and centerunstable manifolds of the departing and arriving points in a Poincaré section. The methodology is applied to obtain the whole set of iso-energetic heteroclinic connections from the center manifold of L 2 to the center manifold of L 1 in the Earth-Moon circular, spatial Restricted Three-Body Problem, for nine increasing energy levels that reach the appearance of Halo orbits in both L 1 and L 2 . Some comments are made on possible applications to space mission design.

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Section: Computing Heteroclinic Connectionsmentioning

confidence: 99%

Semi-analytical computation of heteroclinic connections between center manifolds with the parameterization method

Barcelona¹,

Haro²,

Mondelo³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Finally, DF = DP − I and the iterative procedure (3) can be used to locate limit cycles (we refer the reader to [16] for more details). Once the limit cycle is located, the eigenvalues of the monodromy matrix Dϕ T (x) (x) at the limit cycle, the so-called Floquet characteristic multipliers, give the stability of the limit cycle found.…”

Section: Numerical Computation Of Limit Cycles and Its Stabilitymentioning

confidence: 99%

Saddle-node bifurcation of limit cycles in an epidemic model with two levels of awareness

Juher¹,

Rojas²,

Saldaña³

2022

Preprint

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In this paper we study the appearance of bifurcations of limit cycles in an epidemic model with two types of aware individuals. All the transition rates are constant except for the alerting decay rate of the most aware individuals and the rate of creation of the less aware individuals, which depend on the disease prevalence in a non-linear way. For the ODE model, the numerical computation of the limit cycles and the study of their stability are made by means of the Poincaré map. Moreover, sufficient conditions for the existence of an endemic equilibrium are also obtained. These conditions involve a rather natural relationship between the transmissibility of the disease and that of awareness. Finally, stochastic simulations of the model under a very low rate of imported cases are used to confirm the scenarios of bistability (endemic equilibrium and limit cycle) observed in the solutions of the ODE model.

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“…These periodic orbits are surrounded by orbits with multiple periods called quasi-periodic orbits. Lagrange points and these orbits are invariant manifolds in the sense that their past and future are limited to themselves, and this property is expected to be applicable to spacecraft missions [5]. Lagrange points and periodic orbits have been recognized for a long time and have been the basis for many spacecraft missions.…”

Section: Introductionmentioning

confidence: 99%

Numerical Computation of Quasi-Periodic Solutions in the Circular Restricted Three-Body Problem Based on Percival's Variational Principle

Shibayama,

Miyazaki

2024

Preprint

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There are various solutions to the circular restricted three-body problem, which is used as a model to describe the movement of spacecraft. One of the most famous solutions is the periodic orbits and quasi-periodic solutions that are located near the equilibrium point on the same line as the main and secondary celestial bodies. These solutions are advantageous as they can help reduce fuel consumption while maintaining the spacecraft's orbit. It is expected that these solutions will be applied in many missions. In this paper, we will first explain a functional that uses Percival's variational principle to apply an invariant torus to the circular restricted three-body problem. We will then use a method to obtain an approximate quasi-periodic solution by minimizing this functional using the steepest descent method. Next, we will compare the results obtained from the proposed method with the ones obtained by numerically integrating the circular restricted three-body problem. Finally, we will discuss the prospects of these solutions.

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Computing Invariant Manifolds for Libration Point Missions

Cited by 6 publications

References 32 publications

Semi-analytical computation of heteroclinic connections between center manifolds with the parameterization method

Semi-analytical computation of heteroclinic connections between center manifolds with the parameterization method

Saddle-node bifurcation of limit cycles in an epidemic model with two levels of awareness

Numerical Computation of Quasi-Periodic Solutions in the Circular Restricted Three-Body Problem Based on Percival's Variational Principle

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