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February 2019, 24(2): 563-586. doi: 10.3934/dcdsb.2018197

Efficient representation of invariant manifolds of periodic orbits in the CRTBP

VU University Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands

* Corresponding author: R. Castelli

Received  February 2017 Revised  January 2018 Published  June 2018

This paper deals with a methodology for defining and computing an analytical Fourier-Taylor series parameterisation of local invariant manifolds associated to periodic orbits of polynomial vector fields. Following the Parameterisation Method, the functions involved in the series result by solving some linear non autonomous differential equations. Exploiting the Floquet normal form decomposition, the time dependency is removed and the differential problem is rephrased as an algebraic system to be solved for the Fourier coefficients of the unknown periodic functions. The procedure leads to an efficient and fast computational algorithm. Motivated by mission design purposes, the technique is applied in the framework of the Circular Restricted Three Body problem and the parameterisation of local invariant manifolds for several halo orbits is computed and discussed.

Citation: Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197
References:
[1]

R. L. Anderson and M. W. Lo, Dynamical systems analysis of planetary flybys and approach: Planar europa orbiter, J Guid Control Dynam, 33 (2010), 1899-1912. doi: 10.2514/1.45060.

[2]

K. A. Bokelmann and R. P. Russell, Connecting halo orbits to science orbits at planetary moons, Astrodynamics 2013 - Advances in the Astronautical Sciences: Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, 150 (2014), 1267-1284.

[3]

X. CabréE. Fontich and R. de la Llave, The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328. doi: 10.1512/iumj.2003.52.2245.

[4]

X. CabréE. Fontich and R. de la Llave, The parameterization method for invariant manifolds. Ⅱ. Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), 329-360. doi: 10.1512/iumj.2003.52.2407.

[5]

X. CabréE. Fontich and R. de la Llave, The parameterization method for invariant manifolds. Ⅲ. Overview and applications, J. Differential Equations, 218 (2005), 444-515. doi: 10.1016/j.jde.2004.12.003.

[6]

M. Capiński, Computer assisted existence proofs of Lyapunov orbits at $L_2$ and transversal intersections of invariant manifolds in the Jupiter-Sun PCR3BP, SIAM J. Appl. Dyn. Syst., 11 (2012), 1723-1753. doi: 10.1137/110847366.

[7]

R. Castelli, Regions of prevalence in the coupled restricted three-body problems approximation, Commun Nonlinear Sci Numer Simul, 17 (2012), 804-816. doi: 10.1016/j.cnsns.2011.06.034.

[8]

R. Castelli and J. Lessard, Rigorous numerics in floquet theory: Computing stable and unstable bundles of periodic orbits, SIAM J. Appl. Dyn. Syst., 12 (2013), 204-245. doi: 10.1137/120873960.

[9]

R. CastelliJ. P. Lessard and J. D. Mireles James, Parameterization of invariant manifolds for periodic orbits Ⅰ: Efficient numerics via the floquet normal form, SIAM J. Appl. Dyn. Syst., 14 (2015), 132-167. doi: 10.1137/140960207.

[10]

R. Castelli, J. P. Lessard and J. D. Mireles James, Parameterization of invariant manifolds for periodic orbits Ⅱ: a-posteriori theory and computer assisted error bounds, J. Dyn Diff Equat., (2017).

[11]

R. CastelliJ.-P. Lessard and J.-D. Mireles James, Analytic enclosure of the fundamental matrix solution, Applications of Mathematics, 60 (2015), 617-636. doi: 10.1007/s10492-015-0114-6.

[12]

C. Chicone, Ordinary Differential Equations with Applications, 2$^{nd}$ edition, Springer-Verlag, New York, 2006.

[13]

R. de la Llave and H. Lomelí, Invariant manifolds for analytic difference equations, SIAM J. Appl. Dyn. Syst., 11 (2012), 1614-1651. doi: 10.1137/110858574.

[14]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik, 75 (1997), 293-317. doi: 10.1007/s002110050240.

[15]

M. DellnitzO. JungeM. Post and B. Thiere, On target for Venus - set oriented computation of energy efficient low thrust trajectories, Celest Mech Dyn Astr, 95 (2006), 357-370. doi: 10.1007/s10569-006-9008-y.

[16]

E. Fantino and R. Castelli, Efficient design of direct low-energy transfers in multi-moon systems, Celest Mech Dyn Astr, 127 (2017), 429-450. doi: 10.1007/s10569-016-9733-9.

[17]

J. L. Figueras and A. Haro, Reliable computation of robust response tori on the verge of breakdown, SIAM J. Appl. Dyn. Syst., 11 (2012), 597-628. doi: 10.1137/100809222.

[18]

J. L. FiguerasA. Haro and A. Luque, Rigorous computer assisted applications of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193. doi: 10.1007/s10208-016-9339-3.

[19]

G. Gómez, J. Llibre, C. Simo and R. Martínez, Dynamics and Mission Design Near Libration Points. Vol 1: Fundamentals: The Case of Collinear Libration Points, World Scientific Publishing, 2001. doi: 10.1142/9789812810632_bmatter.

[20]

G. GómezW. S. KoonM. W. LoJ. E. MarsdenJ. Masdemont and S. D. Ross, Connecting orbits and invariant manifolds in the spatial restricted three-body problem, Nonlinearity, 17 (2004), 1571-1606. doi: 10.1088/0951-7715/17/5/002.

[21]

R. GuderM. Dellnitz and E. Kreuzer, An adaptive method for the approximation of the generalized cell mapping, Chaos, Solitons & Fractals, 8 (1997), 525-534. doi: 10.1016/S0960-0779(96)00118-X.

[22]

À. Haro, M. Canadell, A. Luque, J. M. Mondelo and J. L. Figueras, The Parameterization Method for Invariant Manifolds. From Rigorous Results to Effective Computations, Springer International Publishing, 2016. doi: 10.1007/978-3-319-29662-3.

[23]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300. doi: 10.3934/dcdsb.2006.6.1261.

[24]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results, J. Differential Equations, 228 (2006), 530-579. doi: 10.1016/j.jde.2005.10.005.

[25]

K. C. HowellM. BeckmanC. Patterson and D. Folta, Representations of invariant manifolds for applications in three-body systems, Journal of the Astronautical Sciences, 54 (2006), 69-93. doi: 10.1007/BF03256477.

[26]

G. Huguet and R. de la Llave, Computation of limit cycles and their isochrons: Fast algorithms and their convergence, SIAM J. Appl. Dyn. Syst., 12 (2013), 1763-1802. doi: 10.1137/120901210.

[27]

G. HuguetR. de la Llave and Y. Sire, Computation of whiskered invariant tori and their associated manifolds: New fast algorithms, Discrete Contin. Dyn. Syst., 32 (2012), 1309-1353.

[28]

A. Jorba and J. J. Masdemont, Dynamics in the center manifold of the collinear points of the restricted three body problem, Physica D: Nonlinear Phenomena, 132 (1999), 189-213. doi: 10.1016/S0167-2789(99)00042-1.

[29]

W. S. KoonM. W. LoJ. E. Marsden and S. D. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos, 10 (2000), 427-469. doi: 10.1063/1.166509.

[30]

B. Krauskopf and H. Osinga, Computing geodesic level sets on global (un)stable manifolds of vector fields, SIAM J. Appl. Dyn. Syst., 2 (2003), 546-569. doi: 10.1137/030600180.

[31]

B. KrauskopfH. M. OsingaE. J. DoedelM. E. HendersonJ. GuckenheimerA. VladimirskyM. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Int. J. Bifurcation Chaos Appl. Sc. Eng., 15 (2005), 763-791. doi: 10.1142/S0218127405012533.

[32]

H. LeiB. XuX. Hou and Y. Sun, High-order solutions of invariant manifolds associated with libration point orbits in the elliptic restricted three-body system, Celest Mech Dyn Astr, 117 (2013), 349-384. doi: 10.1007/s10569-013-9515-6.

[33]

X. Li and R. de la Llave, Construction of quasi-periodic solutions of delay differential equations via kam techniques, J. Differential Equations, 247 (2009), 822-865. doi: 10.1016/j.jde.2009.03.009.

[34]

A. Luque and J. Villanueva, A KAM theorem without action-angle variables for elliptic lower dimensional tori, Nonlinearity, 24 (2011), 1033-1080. doi: 10.1088/0951-7715/24/4/003.

[35]

J. J. Masdemont, High-order expansions of invariant manifolds of libration point orbits with applications to mission design, Dynamical Systems, 20 (2005), 59-113. doi: 10.1080/14689360412331304291.

[36]

J. D. Mireles James, Fourier-taylor approximation of unstable manifolds for compact maps: Numerical implementation and computer assisted error bounds, Found Comput Math, 17 (2017), 1467-1523. doi: 10.1007/s10208-016-9325-9.

[37]

J. D. Mireles James and H. Lomelí, Computation of heteroclinic arcs with application to the volume preserving Hénon family, SIAM J. Appl. Dyn. Syst., 9 (2010), 919-953. doi: 10.1137/090776329.

[38]

Y. RenJ. J. MasdemontG. Gómez and E. Fantino, Two mechanisms of natural transport in the solar system, Commun Nonlinear Sci Numer Simul, 17 (2012), 844-853. doi: 10.1016/j.cnsns.2011.06.030.

[39]

M. Romero-GómezE. AthanassoulaJ. J. Masdemont and C. García-Gómez, The formation of spiral arms and rings in barred galaxies, A&A, 472 (2007), 63-75.

[40]

C. Simó, Dynamical systems methods for space missions on a vicinity of collinear libration points, Hamiltonian Systems with Three or More Degrees of Freedom (S'Agaró, 1995), 223-241, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, 1999.

[41]

V. Szebehely, Theory of Orbit: The Restricted Problem of Three Bodies, Elsevier Science, 2012.

[42]

M. TantardiniE. FantinoY. RenP. PergolaG. Gómez and J. J. Masdemont, Spacecraft trajectories to the L3 point of the sun-earth three-body problem, Celest Mech Dyn Astr, 108 (2010), 215-232. doi: 10.1007/s10569-010-9299-x.

[43]

F. Topputo, Fast numerical approximation of invariant manifolds in the circular restricted three-body problem, Commun Nonlinear Sci Numer Simul, 32 (2016), 89-98. doi: 10.1016/j.cnsns.2015.08.004.

[44]

J. B. van den BergJ. D. Mireles James and C. Reinhardt, Computing (un)stable manifolds with validated error bounds: non-resonant and resonant spectra, J Nonlinear Sci, 26 (2016), 1055-1095. doi: 10.1007/s00332-016-9298-5.

[45]

J. B. van den BergJ. D. Mireles-JamesJ.-P. Lessard and K. Mischaikow, Rigorous numerics for symmetric connecting orbits: Even homoclinics of the Gray-Scott equation, SIAM Journal on Mathematical Analysis, 43 (2011), 1557-1594. doi: 10.1137/100812008.

[46]

A. ZanzotteraG. MingottiR. Castelli and M. Dellnitz, Intersecting invariant manifolds in spatial restricted three-body problems: Design and optimization of earth-to-halo transfers in the sun-earth-moon scenario, Commun Nonlinear Sci Numer Simul, 17 (2012), 832-843. doi: 10.1016/j.cnsns.2011.06.032.

show all references

References:
[1]

R. L. Anderson and M. W. Lo, Dynamical systems analysis of planetary flybys and approach: Planar europa orbiter, J Guid Control Dynam, 33 (2010), 1899-1912. doi: 10.2514/1.45060.

[2]

K. A. Bokelmann and R. P. Russell, Connecting halo orbits to science orbits at planetary moons, Astrodynamics 2013 - Advances in the Astronautical Sciences: Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, 150 (2014), 1267-1284.

[3]

X. CabréE. Fontich and R. de la Llave, The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328. doi: 10.1512/iumj.2003.52.2245.

[4]

X. CabréE. Fontich and R. de la Llave, The parameterization method for invariant manifolds. Ⅱ. Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), 329-360. doi: 10.1512/iumj.2003.52.2407.

[5]

X. CabréE. Fontich and R. de la Llave, The parameterization method for invariant manifolds. Ⅲ. Overview and applications, J. Differential Equations, 218 (2005), 444-515. doi: 10.1016/j.jde.2004.12.003.

[6]

M. Capiński, Computer assisted existence proofs of Lyapunov orbits at $L_2$ and transversal intersections of invariant manifolds in the Jupiter-Sun PCR3BP, SIAM J. Appl. Dyn. Syst., 11 (2012), 1723-1753. doi: 10.1137/110847366.

[7]

R. Castelli, Regions of prevalence in the coupled restricted three-body problems approximation, Commun Nonlinear Sci Numer Simul, 17 (2012), 804-816. doi: 10.1016/j.cnsns.2011.06.034.

[8]

R. Castelli and J. Lessard, Rigorous numerics in floquet theory: Computing stable and unstable bundles of periodic orbits, SIAM J. Appl. Dyn. Syst., 12 (2013), 204-245. doi: 10.1137/120873960.

[9]

R. CastelliJ. P. Lessard and J. D. Mireles James, Parameterization of invariant manifolds for periodic orbits Ⅰ: Efficient numerics via the floquet normal form, SIAM J. Appl. Dyn. Syst., 14 (2015), 132-167. doi: 10.1137/140960207.

[10]

R. Castelli, J. P. Lessard and J. D. Mireles James, Parameterization of invariant manifolds for periodic orbits Ⅱ: a-posteriori theory and computer assisted error bounds, J. Dyn Diff Equat., (2017).

[11]

R. CastelliJ.-P. Lessard and J.-D. Mireles James, Analytic enclosure of the fundamental matrix solution, Applications of Mathematics, 60 (2015), 617-636. doi: 10.1007/s10492-015-0114-6.

[12]

C. Chicone, Ordinary Differential Equations with Applications, 2$^{nd}$ edition, Springer-Verlag, New York, 2006.

[13]

R. de la Llave and H. Lomelí, Invariant manifolds for analytic difference equations, SIAM J. Appl. Dyn. Syst., 11 (2012), 1614-1651. doi: 10.1137/110858574.

[14]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik, 75 (1997), 293-317. doi: 10.1007/s002110050240.

[15]

M. DellnitzO. JungeM. Post and B. Thiere, On target for Venus - set oriented computation of energy efficient low thrust trajectories, Celest Mech Dyn Astr, 95 (2006), 357-370. doi: 10.1007/s10569-006-9008-y.

[16]

E. Fantino and R. Castelli, Efficient design of direct low-energy transfers in multi-moon systems, Celest Mech Dyn Astr, 127 (2017), 429-450. doi: 10.1007/s10569-016-9733-9.

[17]

J. L. Figueras and A. Haro, Reliable computation of robust response tori on the verge of breakdown, SIAM J. Appl. Dyn. Syst., 11 (2012), 597-628. doi: 10.1137/100809222.

[18]

J. L. FiguerasA. Haro and A. Luque, Rigorous computer assisted applications of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193. doi: 10.1007/s10208-016-9339-3.

[19]

G. Gómez, J. Llibre, C. Simo and R. Martínez, Dynamics and Mission Design Near Libration Points. Vol 1: Fundamentals: The Case of Collinear Libration Points, World Scientific Publishing, 2001. doi: 10.1142/9789812810632_bmatter.

[20]

G. GómezW. S. KoonM. W. LoJ. E. MarsdenJ. Masdemont and S. D. Ross, Connecting orbits and invariant manifolds in the spatial restricted three-body problem, Nonlinearity, 17 (2004), 1571-1606. doi: 10.1088/0951-7715/17/5/002.

[21]

R. GuderM. Dellnitz and E. Kreuzer, An adaptive method for the approximation of the generalized cell mapping, Chaos, Solitons & Fractals, 8 (1997), 525-534. doi: 10.1016/S0960-0779(96)00118-X.

[22]

À. Haro, M. Canadell, A. Luque, J. M. Mondelo and J. L. Figueras, The Parameterization Method for Invariant Manifolds. From Rigorous Results to Effective Computations, Springer International Publishing, 2016. doi: 10.1007/978-3-319-29662-3.

[23]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300. doi: 10.3934/dcdsb.2006.6.1261.

[24]

À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results, J. Differential Equations, 228 (2006), 530-579. doi: 10.1016/j.jde.2005.10.005.

[25]

K. C. HowellM. BeckmanC. Patterson and D. Folta, Representations of invariant manifolds for applications in three-body systems, Journal of the Astronautical Sciences, 54 (2006), 69-93. doi: 10.1007/BF03256477.

[26]

G. Huguet and R. de la Llave, Computation of limit cycles and their isochrons: Fast algorithms and their convergence, SIAM J. Appl. Dyn. Syst., 12 (2013), 1763-1802. doi: 10.1137/120901210.

[27]

G. HuguetR. de la Llave and Y. Sire, Computation of whiskered invariant tori and their associated manifolds: New fast algorithms, Discrete Contin. Dyn. Syst., 32 (2012), 1309-1353.

[28]

A. Jorba and J. J. Masdemont, Dynamics in the center manifold of the collinear points of the restricted three body problem, Physica D: Nonlinear Phenomena, 132 (1999), 189-213. doi: 10.1016/S0167-2789(99)00042-1.

[29]

W. S. KoonM. W. LoJ. E. Marsden and S. D. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos, 10 (2000), 427-469. doi: 10.1063/1.166509.

[30]

B. Krauskopf and H. Osinga, Computing geodesic level sets on global (un)stable manifolds of vector fields, SIAM J. Appl. Dyn. Syst., 2 (2003), 546-569. doi: 10.1137/030600180.

[31]

B. KrauskopfH. M. OsingaE. J. DoedelM. E. HendersonJ. GuckenheimerA. VladimirskyM. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Int. J. Bifurcation Chaos Appl. Sc. Eng., 15 (2005), 763-791. doi: 10.1142/S0218127405012533.

[32]

H. LeiB. XuX. Hou and Y. Sun, High-order solutions of invariant manifolds associated with libration point orbits in the elliptic restricted three-body system, Celest Mech Dyn Astr, 117 (2013), 349-384. doi: 10.1007/s10569-013-9515-6.

[33]

X. Li and R. de la Llave, Construction of quasi-periodic solutions of delay differential equations via kam techniques, J. Differential Equations, 247 (2009), 822-865. doi: 10.1016/j.jde.2009.03.009.

[34]

A. Luque and J. Villanueva, A KAM theorem without action-angle variables for elliptic lower dimensional tori, Nonlinearity, 24 (2011), 1033-1080. doi: 10.1088/0951-7715/24/4/003.

[35]

J. J. Masdemont, High-order expansions of invariant manifolds of libration point orbits with applications to mission design, Dynamical Systems, 20 (2005), 59-113. doi: 10.1080/14689360412331304291.

[36]

J. D. Mireles James, Fourier-taylor approximation of unstable manifolds for compact maps: Numerical implementation and computer assisted error bounds, Found Comput Math, 17 (2017), 1467-1523. doi: 10.1007/s10208-016-9325-9.

[37]

J. D. Mireles James and H. Lomelí, Computation of heteroclinic arcs with application to the volume preserving Hénon family, SIAM J. Appl. Dyn. Syst., 9 (2010), 919-953. doi: 10.1137/090776329.

[38]

Y. RenJ. J. MasdemontG. Gómez and E. Fantino, Two mechanisms of natural transport in the solar system, Commun Nonlinear Sci Numer Simul, 17 (2012), 844-853. doi: 10.1016/j.cnsns.2011.06.030.

[39]

M. Romero-GómezE. AthanassoulaJ. J. Masdemont and C. García-Gómez, The formation of spiral arms and rings in barred galaxies, A&A, 472 (2007), 63-75.

[40]

C. Simó, Dynamical systems methods for space missions on a vicinity of collinear libration points, Hamiltonian Systems with Three or More Degrees of Freedom (S'Agaró, 1995), 223-241, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, 1999.

[41]

V. Szebehely, Theory of Orbit: The Restricted Problem of Three Bodies, Elsevier Science, 2012.

[42]

M. TantardiniE. FantinoY. RenP. PergolaG. Gómez and J. J. Masdemont, Spacecraft trajectories to the L3 point of the sun-earth three-body problem, Celest Mech Dyn Astr, 108 (2010), 215-232. doi: 10.1007/s10569-010-9299-x.

[43]

F. Topputo, Fast numerical approximation of invariant manifolds in the circular restricted three-body problem, Commun Nonlinear Sci Numer Simul, 32 (2016), 89-98. doi: 10.1016/j.cnsns.2015.08.004.

[44]

J. B. van den BergJ. D. Mireles James and C. Reinhardt, Computing (un)stable manifolds with validated error bounds: non-resonant and resonant spectra, J Nonlinear Sci, 26 (2016), 1055-1095. doi: 10.1007/s00332-016-9298-5.

[45]

J. B. van den BergJ. D. Mireles-JamesJ.-P. Lessard and K. Mischaikow, Rigorous numerics for symmetric connecting orbits: Even homoclinics of the Gray-Scott equation, SIAM Journal on Mathematical Analysis, 43 (2011), 1557-1594. doi: 10.1137/100812008.

[46]

A. ZanzotteraG. MingottiR. Castelli and M. Dellnitz, Intersecting invariant manifolds in spatial restricted three-body problems: Design and optimization of earth-to-halo transfers in the sun-earth-moon scenario, Commun Nonlinear Sci Numer Simul, 17 (2012), 832-843. doi: 10.1016/j.cnsns.2011.06.032.

Figure 1.  Schematic picture of the action of the conjugating map. The function $\mathbb{P}(\theta, \sigma)$ maps the cylinder in parameter space into the local manifold, in such a way that trajectories of the linear vector field (5) are mapped into trajectories on the manifold.
Figure 2.  Image of the parameterisation of the local stable (left) and unstable (right) manifold associated to the periodic orbit of Example 1. The black line is the periodic orbit.
Figure 3.  Image of the parameterisation of the manifolds associated to the Lyapunov orbit of Example 2. The figures at the bottom show the image of $\mathbb{P}(\theta, \sigma)$ for some fixed values of $\theta$ and of $\sigma$.
Figure 4.  Unstable local invariant manifold for the Halo orbit of example 3.
Figure 5.  Left: values of the $\log(|a_{\alpha, k}|)$ for $\alpha = 1$ (top line) up to $\alpha = 10$ (bottom line). Right: plot of $\|a_\alpha\|$ in logarithmic scale.
Figure 6.  Image of the parameterisation of the stable and unstable local manifold associated to the orbit of Exemple 6.
Figure 7.  Left: The blu-green figure is the image of the parameterisation for the local unstable manifold, the red lines are fibres on the manifold obtained by numerical integration. Right: Visualisation of the planes $\Pi_1$, $\Pi_2$.
Figure 8.  Poincaré sections of the manifolds at the planes $\Pi_1$, $\Pi_2$. The blue line is the section of the manifold obtained with the parameterisation, the red dots refer to the manifold obtained by numerical integration.
Figure 9.  Heteroclinic obit. The black curves are the periodic Lyapunov orbits, the red circles are the sections on the stable and unstable manifolds. The green line is the curve connecting the two manifolds, obtained by solving $F(\theta_1, \theta_2, T) = 0$. The blue lines are orbits on the stable and unstable manifolds.
Table 1.  Computational time
m N time (s)
30 6 1.2
30 10 8.6
40 6 1.4
40 10 9.1
50 4 0.4
50 6 1.9
50 9 7.6
m N time (s)
30 6 1.2
30 10 8.6
40 6 1.4
40 10 9.1
50 4 0.4
50 6 1.9
50 9 7.6
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