Lagrangian coherent structures in the planar elliptic restricted three-body problem

Gawlik, Evan S.; Marsden, Jerrold E.; Toit, Philip C. Du; Campagnola, Stefano

doi:10.1007/s10569-008-9180-3

Cited by 66 publications

(50 citation statements)

References 18 publications

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“…With the appropriate visualization tools, computations in these phase spaces can aid the search for dynamical structure in mechanical and biomechanical data from experiments or computational models, such as separatrices between stable and unstable motions. 44,15,49,58,59,50 Additionally, it may be of interest to detect LCS structure on highdimensional invariant manifolds of possibly high curvature, which are commonly found in phase space, such as energy manifolds, center manifolds, and stable and unstable manifolds.…”

Section: Discussionmentioning

confidence: 99%

The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds

Lekien

Ross

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

118

135

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We generalize the concepts of finite-time Lyapunov exponent ͑FTLE͒ and Lagrangian coherent structures to arbitrary Riemannian manifolds. The methods are illustrated for convection cells on cylinders and Möbius strips, as well as for the splitting of the Antarctic polar vortex in the spherical stratosphere and a related point vortex model. We modify the FTLE computational method and accommodate unstructured meshes of triangles and tetrahedra to fit manifolds of arbitrary shape, as well as to facilitate dynamic refinement of the FTLE mesh. © 2010 American Institute of Physics. ͓doi:10.1063/1.3278516͔Riemannian manifolds are ubiquitous in science and engineering, being the more natural mathematical setting for many dynamical systems. For instance, transport along isopycnal surfaces in the ocean and large-scale mixing in the atmosphere are processes taking place on a curved manifold, not a vector space. In this paper, we generalize the notion of finite-time Lyapunov exponent (FTLE) and Lagrangian coherent structures (LCS) to arbitrary Riemannian differentiable manifolds. We show that both notions are independent of the coordinate system but depend on the chosen metric. However, we find that LCS do not depend much on the metric. The FTLE measures separation and tends to be large and positive along LCS and very small elsewhere. For sufficiently large integration times, the steep variations of the FTLE field cannot be modified much by smooth changes in the metric and the LCS remain essentially unchanged. Approximating, or even ignoring, the manifold metric does not influence the result for large integration times, and therefore, computing the FTLE field on manifolds is robust. Aside from these conclusions, we present a general algorithm for computing the FTLE on manifolds covered with meshes of polyhedra. The algorithm requires knowledge of the mesh nodes, the image of the mesh nodes under the flow, as well as information about node neighbors (but not the full connectivity). We also used the same algorithm in Euclidian spaces where the unstructured mesh permits efficient adaptive refinement for capturing sharp LCS features. We illustrate the results and methods on several systems: convection cells in a plane, on a cylinder, and on a Möbius strip, as well as atmospheric transport resulting from the 2002 splitting of the Antarctic ozone hole in the spherical stratosphere and a related point vortex model on the sphere.

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Section: Discussionmentioning

confidence: 99%

The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds

Lekien

Ross

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

118

135

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“…For instance in fluid dynamics [10,14,16] or astrodynamics [17,4]. The main use of the LCS is to determine and distinguish different regions of the phase space as a function of the behaviour of the orbits inside each one.…”

Section: Introductionmentioning

confidence: 99%

Tools to detect structures in dynamical systems using Jet Transport

Pérez-Palau

Masdemont

Gómez

2015

Celest Mech Dyn Astr

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This paper is devoted to the development of some dynamical indicators that allow the determination of regions and structures that separate different dynamic regimes in autonomous and non-autonomous dynamical systems. The underlying idea is closely related to the Lagrangian Coherent Structures concept introduced by Haller. In the present paper, instead of using the Cauchy-Green tensor, that determines the domains where the flow associated to a differential equation is expanding in the normal direction, the Jet Transport methodology is used. This is a semi-numerical tool, that has as basic ingredients a polynomial algebra package and a numerical integration method, allowing, at each integration step, the propagation under a flow of a neighbourhood U instead of a single initial condition. The output of the procedure is a polynomial in several variables that represents the image of U up to a selected order, containing high order terms of the variational equations. Using these high order representation, the places where the normal direction expands can be easily detected, in a similar manner as the procedures for calculating the Lagrangian Coherent Structures do. In order to illustrate the methodology, first the results obtained in the determination of the separatrices of the simple and the periodically perturbed pendulum are given. Later, the applications to the circular restricted three body problem are considered, where the aim is the detection of invariant manifolds of libration point orbits, as well as in the non-autonomous vector field defined by the elliptic restricted three body problem.

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“…These problems are described by non-autonomous Hamiltonian systems where the notions of stable and unstable manifolds are not well-defined in the phase space 1 . It seems possible that the WSB may turn out to provide a good substitute for the hyperbolic invariant manifolds in such models (see [23]). …”

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confidence: 99%

Weak Stability Boundary and Invariant Manifolds

Belbruno¹,

Gidea²,

Topputo³

2010

SIAM J. Appl. Dyn. Syst.

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Abstract. The concept of weak stability boundary has been successfully used in the design of several fuel efficient space missions. In this paper we give a rigorous definition of the weak stability boundary in the context of the planar circular restricted three-body problem, and we provide a geometric argument for the fact that, for some energy range, the points in the weak stability boundary of the small primary are the points with zero radial velocity that lie on the stable manifolds of the Lyapunov orbits about the libration points L 1 and L 2 , provided that these manifolds satisfy some topological conditions. The geometric method is based on the property of the invariant manifolds of Lyapunov orbits being separatrices of the energy manifold. We support our geometric argument with numerical experiments.Key words. Weak Stability Boundary, Lyapunov Orbits, Invariant Manifolds AMS subject classifications. 70F15, 70F07, 37D101. Introduction. The notion of weak stability boundary (WSB) was first introduced heuristically by Belbruno in 1987 for designing fuel efficient space missions and was proven to be useful in applications [2,3,4,12,13,6,45,33,14,35,22,7,39,38,44]. The WSB can be used to construct low energy transfers to the Moon, requiring little or no fuel for capture into lunar orbit. The first application for an operational spacecraft occurred in 1991 with the rescue of the Japanese mission Hiten. WSB was also applied in ESA's spacecraft SMART-1 in 2004 (see [40]). The WSB technique will be applied again in ESA's mission BepiColombo to explore planet Mercury in 2013 (see [25]), and in some upcoming NASA missions.A different methodology of designing fuel efficient trajectories, based on hyperbolic invariant manifolds, was proposed in [1,26,28,29,30,31], and was successfully applied in several space missions (see also [18,19,20,21]). In some of these works it has been suggested that the hyperbolic invariant manifold method can be used to explain the trajectories obtained through the WSB method. Supporting this assertion, García and Gómez present in [22] numerical explorations that suggests that, for some range of energies, the WSB is contained in the closure of the union of the stable manifolds of the periodic orbits about two of the equilibrium points of the planar circular restricted three-body problem.In this paper we use the separatrix property of the invariant manifolds of the periodic orbits about the equilibrium points to argue that, under some conditions, the points on the stable manifolds are WSB points. We support our geometrical argument with numerical experiments. Our result, corroborated with the result in [22], demonstrates, by double inclusion, that, for some range of energies, the WSB coincides with the set of points on the stable manifolds with zero radial velocity and negative Kepler energy relative to the small primary.In Section 2 we give some background on the planar circular restricted three-body problem. In Section 3 we define the WSB as follows: in the context of the planar circular r...

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Lagrangian coherent structures in the planar elliptic restricted three-body problem

Cited by 66 publications

References 18 publications

The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds

The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds

Tools to detect structures in dynamical systems using Jet Transport

Weak Stability Boundary and Invariant Manifolds

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