DSpace Community:
http://hdl.handle.net/2117/3227
Mon, 21 Apr 2014 10:24:44 GMT20140421T10:24:44Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoCenters of quasihomogeneous polynomial differential equations of degree three
http://hdl.handle.net/2117/22462
Title: Centers of quasihomogeneous polynomial differential equations of degree three
Authors: Aziz, Waleed; Llibre Saló, Jaume; Pantazi, Chara
Abstract: We characterize the centers of the quasihomogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first orderMon, 31 Mar 2014 16:38:52 GMThttp://hdl.handle.net/2117/2246220140331T16:38:52ZAziz, Waleed; Llibre Saló, Jaume; Pantazi, CharanoQuasihomogeneous polynomial systems, Centers, Limit cycles, cubic vectorfields, plane, flowsWe characterize the centers of the quasihomogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first orderOn the cuckersmale flocking model applied to a formation moving in a central force field around the Earth
http://hdl.handle.net/2117/22420
Title: On the cuckersmale flocking model applied to a formation moving in a central force field around the Earth
Authors: Paita, Fabrizio; Gomez Muntaner, Gerard; Masdemont Soler, Josep
Abstract: The Cucker – Smale (CS) flocking model is an interacting particles control system where every particle adjusts
its dynamics according to a weighted average between its velocity and those of the other elements of the flock.
Under this model, a rigid body configuration can be achieved exponentially fast for suitable initial configurations.
Furthermore, as shown by J. Shen, if the dynamics of the formation is driven by the presence of a free willed leader
then similar asymptotic results can be obtained. In the present paper we extend the CS control law in the context of a
multi – spacecraft system moving around a central body. Some new analytical results are given, as well as the results
of the numerical explorations that have been done to evaluate the performances of the control law, with particular
attention to its interaction with a central gravitational field. These performances have been evaluated mainly through
the use of two indicators: the evolution of the relative spacecrafts distances and the total fuel expended to maintain
the formation.Thu, 27 Mar 2014 19:10:58 GMThttp://hdl.handle.net/2117/2242020140327T19:10:58ZPaita, Fabrizio; Gomez Muntaner, Gerard; Masdemont Soler, JosepnoThe Cucker – Smale (CS) flocking model is an interacting particles control system where every particle adjusts
its dynamics according to a weighted average between its velocity and those of the other elements of the flock.
Under this model, a rigid body configuration can be achieved exponentially fast for suitable initial configurations.
Furthermore, as shown by J. Shen, if the dynamics of the formation is driven by the presence of a free willed leader
then similar asymptotic results can be obtained. In the present paper we extend the CS control law in the context of a
multi – spacecraft system moving around a central body. Some new analytical results are given, as well as the results
of the numerical explorations that have been done to evaluate the performances of the control law, with particular
attention to its interaction with a central gravitational field. These performances have been evaluated mainly through
the use of two indicators: the evolution of the relative spacecrafts distances and the total fuel expended to maintain
the formation.Mathematical Methods Applied to the Celestial Mechanics of Artificial Satellites 2013
http://hdl.handle.net/2117/22393
Title: Mathematical Methods Applied to the Celestial Mechanics of Artificial Satellites 2013
Authors: Prado, Antonio F. Bertachini A.; Masdemont Soler, Josep; Zanardi, Maria Cecilia; Winter, Silvia Maria Giuliatti; Yokoyama, Tadashi; Gomes, Vivian MartinsWed, 26 Mar 2014 15:17:42 GMThttp://hdl.handle.net/2117/2239320140326T15:17:42ZPrado, Antonio F. Bertachini A.; Masdemont Soler, Josep; Zanardi, Maria Cecilia; Winter, Silvia Maria Giuliatti; Yokoyama, Tadashi; Gomes, Vivian MartinsnoComputation of limit cycles and their isochrons: Fast algorithms and their convergence
http://hdl.handle.net/2117/22392
Title: Computation of limit cycles and their isochrons: Fast algorithms and their convergence
Authors: Huguet Casades, Gemma; de la Llave Canosa, Rafael
Abstract: We present efficient algorithms to compute limit cycles and their isochrons (i.e., the sets of points with the same asymptotic phase) for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrons, and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation of the Newton step efficiently. The algorithms are efficient in the sense that if we discretize the functions using N points, a Newton step requires O(N) storage and O(N log N) operations in Fourier discretization or O(N) operations in other discretizations. We prove convergence of the algorithms and present a validation theorem in an a posteriori format. That is, we show that if there is an approximate solution of the invariance equation that satisfies some some mild nondegeneracy conditions, then there is a true solution nearby. Thus, our main theorem can be used to validate numerically computed solutions. The theorem also shows that the isochrons are analytic and depend analytically on the base point. Moreover, it establishes smooth dependence of the solutions on parameters and provides efficient algorithms to compute perturbative expansions with respect to external parameters. We include a discussion on the numerical implementation of the algorithms as well as numerical results for representative examples.Wed, 26 Mar 2014 15:06:49 GMThttp://hdl.handle.net/2117/2239220140326T15:06:49ZHuguet Casades, Gemma; de la Llave Canosa, Rafaelnoparameterization method, isochrons, invariant manifolds, numerical computation of invariant objects, phase resetting curves, biological oscillatorsWe present efficient algorithms to compute limit cycles and their isochrons (i.e., the sets of points with the same asymptotic phase) for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrons, and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation of the Newton step efficiently. The algorithms are efficient in the sense that if we discretize the functions using N points, a Newton step requires O(N) storage and O(N log N) operations in Fourier discretization or O(N) operations in other discretizations. We prove convergence of the algorithms and present a validation theorem in an a posteriori format. That is, we show that if there is an approximate solution of the invariance equation that satisfies some some mild nondegeneracy conditions, then there is a true solution nearby. Thus, our main theorem can be used to validate numerically computed solutions. The theorem also shows that the isochrons are analytic and depend analytically on the base point. Moreover, it establishes smooth dependence of the solutions on parameters and provides efficient algorithms to compute perturbative expansions with respect to external parameters. We include a discussion on the numerical implementation of the algorithms as well as numerical results for representative examples.Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
http://hdl.handle.net/2117/22391
Title: Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
Authors: Cabré Vilagut, Xavier; Sire, Yannick
Abstract: This is the first of two articles dealing with the equation ()sv = f (v) in Rn, with s ¿ (0,1), where ()s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in Rn+1+ together with a nonlinear Neumann boundary condition on ¿Rn+1 + =Rn.
In this ¿rst article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a followup paper to be also suficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform ass ¿1, establishing in the limit the corresponding known results for the Laplacian.
In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.http://hdl.handle.net/2117/22391Cabré Vilagut, Xavier; Sire, YannicknoThis is the first of two articles dealing with the equation ()sv = f (v) in Rn, with s ¿ (0,1), where ()s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in Rn+1+ together with a nonlinear Neumann boundary condition on ¿Rn+1 + =Rn.
In this ¿rst article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a followup paper to be also suficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform ass ¿1, establishing in the limit the corresponding known results for the Laplacian.
In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.A Perturbation argument for a Monge–Ampère type equation arising in optimal transportation
http://hdl.handle.net/2117/22385
Title: A Perturbation argument for a Monge–Ampère type equation arising in optimal transportation
Authors: Caffarelli, Luis; González Nogueras, María del Mar; Nguyen, Truyen
Abstract: We prove some interior regularity results for potential functions of optimal transportation problems with power costs. The main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost, but it does not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard Monge–Ampère equation in order to obtain, first, interior C1,1 estimates for the potential and, second, interior Hölder estimates for
second derivatives. In particular, we take a close look at the geometry of optimal
transportation when the cost function is close to quadratic in order to understand
how the equation degenerates near the boundary.Wed, 26 Mar 2014 07:03:34 GMThttp://hdl.handle.net/2117/2238520140326T07:03:34ZCaffarelli, Luis; González Nogueras, María del Mar; Nguyen, TruyennoWe prove some interior regularity results for potential functions of optimal transportation problems with power costs. The main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost, but it does not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard Monge–Ampère equation in order to obtain, first, interior C1,1 estimates for the potential and, second, interior Hölder estimates for
second derivatives. In particular, we take a close look at the geometry of optimal
transportation when the cost function is close to quadratic in order to understand
how the equation degenerates near the boundary.Generalized CliffordSeveri inequality and the volume of irregular varieties
http://hdl.handle.net/2117/22384
Title: Generalized CliffordSeveri inequality and the volume of irregular varieties
Authors: Barja Yáñez, Miguel Ángel
Abstract: We give a sharp lower bound for the selfintersection of a nef li
ne bundle L on an irregular variety X in terms of its continuous global sections and the Albanese dim
ension of X, which we call the Generalized CliffordSeveri inequality. We also extend the result to nef
vector bundles and give a slope inequality for fibred irregular varieties. As a byproduct we obtain a lower b
ound for the volume of irregular varieties; when X is of maximal Albanese dimension the bound is vol(X)=2n!¿¿X
and it is sharp.
Description: Preprint. Acceptat per publicar a Duke Math. J.Tue, 25 Mar 2014 20:15:26 GMThttp://hdl.handle.net/2117/2238420140325T20:15:26ZBarja Yáñez, Miguel ÁngelnoSeveri inequality
Slope
maximal Albanese Varieties
VolumeWe give a sharp lower bound for the selfintersection of a nef li
ne bundle L on an irregular variety X in terms of its continuous global sections and the Albanese dim
ension of X, which we call the Generalized CliffordSeveri inequality. We also extend the result to nef
vector bundles and give a slope inequality for fibred irregular varieties. As a byproduct we obtain a lower b
ound for the volume of irregular varieties; when X is of maximal Albanese dimension the bound is vol(X)=2n!¿¿X
and it is sharp.Local description of phylogenetic groupbased models
http://hdl.handle.net/2117/22383
Title: Local description of phylogenetic groupbased models
Authors: Casanellas Rius, Marta; Fernández Sánchez, Jesús; Michalek, Mateusz
Abstract: Motivated by phylogenetics, our aim is to obtain a system of equations that
de ne a phylogenetic variety on an open set containing the biologically meaningful points. In
this paper we consider phylogenetic varieties de ned via groupbased models. For any nite
abelian group G, we provide an explicit construction of codimX phylogenetic invariants
(polynomial equations) of degree at most jGj that de ne the variety X on a Zariski open set
U. The set U contains all biologically meaningful points when G is the group of the Kimura
3parameter model. In particular, our main result con rms [Mic12, Conjecture 7.9] and, on
the set U, Conjectures 29 and 30 of [SS05].Tue, 25 Mar 2014 19:26:25 GMThttp://hdl.handle.net/2117/2238320140325T19:26:25ZCasanellas Rius, Marta; Fernández Sánchez, Jesús; Michalek, Mateusznogroupbased model, phylogenetic invariant, toric varietyMotivated by phylogenetics, our aim is to obtain a system of equations that
de ne a phylogenetic variety on an open set containing the biologically meaningful points. In
this paper we consider phylogenetic varieties de ned via groupbased models. For any nite
abelian group G, we provide an explicit construction of codimX phylogenetic invariants
(polynomial equations) of degree at most jGj that de ne the variety X on a Zariski open set
U. The set U contains all biologically meaningful points when G is the group of the Kimura
3parameter model. In particular, our main result con rms [Mic12, Conjecture 7.9] and, on
the set U, Conjectures 29 and 30 of [SS05].Testing of the three multiplicatively closed (Lie Markov) model heirarchies which respect purine/pyrimidine, WatsonCrick, and amino/keto nucleotide groupings
http://hdl.handle.net/2117/22382
Title: Testing of the three multiplicatively closed (Lie Markov) model heirarchies which respect purine/pyrimidine, WatsonCrick, and amino/keto nucleotide groupings
Authors: Woodhams, Michael D.; Fernández Sánchez, Jesús; Sumner, Jeremy
Abstract: We present three hierarchies of Lie Markov models of DNA sequence evolution. These models are
(locally) “multiplicatively closed,” meaning that the composition of two Markov matrices in the
model results, with some (rare) exceptions, in a third Markov matrix that is still in the model.
Additionally, the models in each hierarchy respectively distinguish between (i) purines and pyrimadines
(RY), (ii) WatsonCrick pairs (WS), and (iii) amino/keto pairs (MK), but otherwise treat
the four nucleotides without distinction. The multiplicative closure property allows mathematically
consistent modeling of timeinhomogeneous scenarios, unlike models that are not closed, such
as the general timereversible model (GTR) and many of its submodels. We derive the nesting
relationships of the three model hierarchies and present software implementing the models. For a
diverse range of biological data sets, we perform Bayesian information criterion model comparision
analogous to that of the ModelTest framework. We find that our models outperform the GTR
model in some (but not all) cases.Tue, 25 Mar 2014 17:59:41 GMThttp://hdl.handle.net/2117/2238220140325T17:59:41ZWoodhams, Michael D.; Fernández Sánchez, Jesús; Sumner, JeremynoLie Markov models, multiplicative closure, DNA evolutionWe present three hierarchies of Lie Markov models of DNA sequence evolution. These models are
(locally) “multiplicatively closed,” meaning that the composition of two Markov matrices in the
model results, with some (rare) exceptions, in a third Markov matrix that is still in the model.
Additionally, the models in each hierarchy respectively distinguish between (i) purines and pyrimadines
(RY), (ii) WatsonCrick pairs (WS), and (iii) amino/keto pairs (MK), but otherwise treat
the four nucleotides without distinction. The multiplicative closure property allows mathematically
consistent modeling of timeinhomogeneous scenarios, unlike models that are not closed, such
as the general timereversible model (GTR) and many of its submodels. We derive the nesting
relationships of the three model hierarchies and present software implementing the models. For a
diverse range of biological data sets, we perform Bayesian information criterion model comparision
analogous to that of the ModelTest framework. We find that our models outperform the GTR
model in some (but not all) cases.Lie Markov models with purine/pyrimidine symmetry
http://hdl.handle.net/2117/22381
Title: Lie Markov models with purine/pyrimidine symmetry
Authors: Fernández Sánchez, Jesús; Sumner, Jeremy; Jarvis, Peter; Woodhams, Michael D.
Abstract: Continuoustime Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying timeindependent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with timedependent rates satisfying the same constraints. It is then useful to require that there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of "Lie Markov models", which are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines  that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally defined over the complex field, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra.Tue, 25 Mar 2014 17:13:58 GMThttp://hdl.handle.net/2117/2238120140325T17:13:58ZFernández Sánchez, Jesús; Sumner, Jeremy; Jarvis, Peter; Woodhams, Michael D.noevolutionary model, group representation theory, Lie algebraContinuoustime Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying timeindependent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with timedependent rates satisfying the same constraints. It is then useful to require that there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of "Lie Markov models", which are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines  that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally defined over the complex field, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra.Estructuras Ainfinito en la opérada de cactus
http://hdl.handle.net/2117/22097
Title: Estructuras Ainfinito en la opérada de cactus
Authors: Gálvez Carrillo, Maria Immaculada; Lombardi, Leandro; Tonks, Andrew
Abstract: Diversas versiones de la opérada de cactus inicialmente definida por Voronov han sido estudiadas. Es conocida su equivalencia débil con la opérada de pequeños discos. Se conoce pues que la opérada de cactus admite una acción de la opérada de Gerstenhaber salvo homotopía. En este proyecto, nuestro objetivo es obtener una realización explícita de dicha acción. Por el momento, hemos construido una acción explícita de la opérada A8 en la opérada de cactus, que presentamos en este pósterhttp://hdl.handle.net/2117/22097Gálvez Carrillo, Maria Immaculada; Lombardi, Leandro; Tonks, AndrewnoDiversas versiones de la opérada de cactus inicialmente definida por Voronov han sido estudiadas. Es conocida su equivalencia débil con la opérada de pequeños discos. Se conoce pues que la opérada de cactus admite una acción de la opérada de Gerstenhaber salvo homotopía. En este proyecto, nuestro objetivo es obtener una realización explícita de dicha acción. Por el momento, hemos construido una acción explícita de la opérada A8 en la opérada de cactus, que presentamos en este pósterGroupoids and Faà di Bruno formulae for Green functions in bialgebras of trees
http://hdl.handle.net/2117/22088
Title: Groupoids and Faà di Bruno formulae for Green functions in bialgebras of trees
Authors: Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew
Abstract: We prove a Faà di Bruno formula for the Green function in the bialgebra of Ptrees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids.Mon, 17 Mar 2014 09:08:32 GMThttp://hdl.handle.net/2117/2208820140317T09:08:32ZGálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, AndrewnoTrees, Groupoids, Homotopy cardinality, Polynomial functors, Bialgebras, Perturbative methods of renormalisationWe prove a Faà di Bruno formula for the Green function in the bialgebra of Ptrees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids.Jet Transport propagation of uncertainties for orbits around the Earth
http://hdl.handle.net/2117/21808
Title: Jet Transport propagation of uncertainties for orbits around the Earth
Authors: Pérez, Daniel; Masdemont Soler, Josep; Gómez Muntané, Gerard
Abstract: In this paper we present a tool to study the nonlinear propagation of uncertainties for orbits around the Earth.
The tool we introduce is known as Jet Transport and allows to propagate full neighborhoods of initial states instead of
a single initial state by means of usual numerical integrators. The description of the transported neighborhood is ob
tained in a semianalytical way by means of polynomials in 6 variables. These variables correspond to displacements
in the phase space from the reference point selected in an orbit as initial condition. The basis of the procedure is a
standard numerical integrator of ordinary differential equations (such as a RungeKutta or a Taylor method) where
the usual arithmetic is replaced by a polynomial arithmetic. In this way, the solution of the variational equations is
also obtained up to high order. This methodology is applied to the propagation of satellite trajectories and to the
computation of images of uncertainty ellipsoids including high order nonlinear descriptions. The procedure can be
specially adapted to the determination of collision probabilities with catalogued space debris or for the end of life
analysis of spacecraft in Medium Earth OrbitsFri, 28 Feb 2014 11:24:37 GMThttp://hdl.handle.net/2117/2180820140228T11:24:37ZPérez, Daniel; Masdemont Soler, Josep; Gómez Muntané, GerardnoIn this paper we present a tool to study the nonlinear propagation of uncertainties for orbits around the Earth.
The tool we introduce is known as Jet Transport and allows to propagate full neighborhoods of initial states instead of
a single initial state by means of usual numerical integrators. The description of the transported neighborhood is ob
tained in a semianalytical way by means of polynomials in 6 variables. These variables correspond to displacements
in the phase space from the reference point selected in an orbit as initial condition. The basis of the procedure is a
standard numerical integrator of ordinary differential equations (such as a RungeKutta or a Taylor method) where
the usual arithmetic is replaced by a polynomial arithmetic. In this way, the solution of the variational equations is
also obtained up to high order. This methodology is applied to the propagation of satellite trajectories and to the
computation of images of uncertainty ellipsoids including high order nonlinear descriptions. The procedure can be
specially adapted to the determination of collision probabilities with catalogued space debris or for the end of life
analysis of spacecraft in Medium Earth OrbitsEndoflife disposal of libration point orbit spacecraft
http://hdl.handle.net/2117/21807
Title: Endoflife disposal of libration point orbit spacecraft
Authors: Olikara, Zubin; Gómez Muntané, Gerard; Masdemont Soler, Josep
Abstract: In this work we investigate endoflife trajectories for
spacecraft in orbit about the SunEarth
L
1
and
L
2
libration points. A plan for decommission is often re
quired during the mission design process. We study
the spacecraft's natural dynamics in both a high
delity model and the circular restricted threebody
problem. In particular, we consider the role of the
unstable manifold and forbidden regions in determin
ing disposal outcomes. A simple maneuver scheme
to prevent returns to the Earth vicinity is also an
alyzed. We include discussion on potential collision
orbit schemes.Fri, 28 Feb 2014 11:10:49 GMThttp://hdl.handle.net/2117/2180720140228T11:10:49ZOlikara, Zubin; Gómez Muntané, Gerard; Masdemont Soler, JosepnoIn this work we investigate endoflife trajectories for
spacecraft in orbit about the SunEarth
L
1
and
L
2
libration points. A plan for decommission is often re
quired during the mission design process. We study
the spacecraft's natural dynamics in both a high
delity model and the circular restricted threebody
problem. In particular, we consider the role of the
unstable manifold and forbidden regions in determin
ing disposal outcomes. A simple maneuver scheme
to prevent returns to the Earth vicinity is also an
alyzed. We include discussion on potential collision
orbit schemes.A note on the dynamics around the Lagrange points of the EarthMoon system in a complete Solar System model
http://hdl.handle.net/2117/21806
Title: A note on the dynamics around the Lagrange points of the EarthMoon system in a complete Solar System model
Authors: Lian, Yijun; Gómez Muntané, Gerard; Masdemont Soler, Josep; Tang, Guojian
Abstract: This paper studies the dynamics of a massless particle around the
libration points of the Earth

Moon system in a
full Solar System gravitational
model. The study is based on the analysis of the quasi

periodic solutions around the
equilibrium points. For the analysis and computation of the quasi

periodic orbits, a novel iterative algorithm is
introduced which is
a combination of
the
multiple shooti
ng method
and
a refined Fourier analysis of the orbits
computed with the multiple shooting. Using as initial seeds the libration point orbits of Circular Restricted Three
Body Problem, determined by Lindstedt

Poincar
é
methods, the procedure is able to refi
ne them in a complete Solar
System model for large time

spans covering most of the relevant Sun

Earth

Moon periods. For the collinear points,
the developed approach works well and reveals the strong relevance of the phase space around the equilibrium point
s
in both models. For the triangular points, difficulties appear and an intermediate model (bicircular model, BCM) is
introduced to aid the refinement.Fri, 28 Feb 2014 10:59:44 GMThttp://hdl.handle.net/2117/2180620140228T10:59:44ZLian, Yijun; Gómez Muntané, Gerard; Masdemont Soler, Josep; Tang, GuojiannoThis paper studies the dynamics of a massless particle around the
libration points of the Earth

Moon system in a
full Solar System gravitational
model. The study is based on the analysis of the quasi

periodic solutions around the
equilibrium points. For the analysis and computation of the quasi

periodic orbits, a novel iterative algorithm is
introduced which is
a combination of
the
multiple shooti
ng method
and
a refined Fourier analysis of the orbits
computed with the multiple shooting. Using as initial seeds the libration point orbits of Circular Restricted Three
Body Problem, determined by Lindstedt

Poincar
é
methods, the procedure is able to refi
ne them in a complete Solar
System model for large time

spans covering most of the relevant Sun

Earth

Moon periods. For the collinear points,
the developed approach works well and reveals the strong relevance of the phase space around the equilibrium point
s
in both models. For the triangular points, difficulties appear and an intermediate model (bicircular model, BCM) is
introduced to aid the refinement.