Celestial Mechanics and Dynamical Astronomy

, Volume 123, Issue 3, pp 239–262 | Cite as

Tools to detect structures in dynamical systems using Jet Transport

  • Daniel Pérez-PalauEmail author
  • Josep J. Masdemont
  • Gerard Gómez
Original Article


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.


Lagrangian coherent structures (LCS) Jet Transport Invariant structures Simple pendulum Perturbed pendulum Circular restricted three-body problem Elliptic restricted three-body problem Finite time Lyapunov exponents 



The authors would like to thank to the anonymous reviewers for their comments and fruitful suggestions to improve the quality of the paper. This work has been supported by the Spanish Grants MTM2010-16425, MTM2013-41168-P (G.G.), MTM2012-31714, 2014SGR504 (J.J.M.), and MTM-2010-16425, MTM2013-41168-P, AP2010-0268 (D.P.).


  1. Abad, A., Barrio, R., Blesa, F., Rodríguez, M.: Algorithm924: TIDES, a taylor series integrator for differential equations. ACM Trans. Math. Softw. 39(1), 1–28 (2012)CrossRefGoogle Scholar
  2. Alessi, E.M., Farrès, A., Vieiro, A., Jorba, À., Simó, C.: Jet Transport and applications to neo’s. In: Proceedings of 1st IAA Planetary Defense Conference. ESA Conference Bureau, Granada, Spain (2009)Google Scholar
  3. Armellin, R., Di Lizia, P., Berenlli-Zazzera, F., Berz, M.: Asteroid close encounters characterization using differential algebra: the case of apophis. Celest. Mech. Dyn. Astron. 107(4), 451–470 (2010)zbMATHCrossRefADSGoogle Scholar
  4. Gawlik, E.S., Marsden, J.E., Du Toit, P.C., Campagnola, S.: Lagrangian coherent structures in the planar elliptic restricted three-body problem. Celest. Mech. Dyn. Astron. 103(3), 227–249 (2009)zbMATHCrossRefADSGoogle Scholar
  5. Gómez, G., Jorba, À., Masdemont, J.J., Simó, C.: Dynamics and Mission Design Near Libration Points. Vol. IV Advanced Methods for Triangular Points. World Scientific, Singapore (2001)Google Scholar
  6. Haller, G.: A variational theory of hyperbolic Lagrangian coherent structures. Phys. D Nonlinear Phenom. 40(7), 574–598 (2011)MathSciNetCrossRefADSGoogle Scholar
  7. Haller, G., Yuan, G.: Lagrangian coherent structures and mixing in two-dimensional turbulence. Phys. D Nonlinear Phenom. 147(3–4), 352–370 (2000)zbMATHMathSciNetCrossRefADSGoogle Scholar
  8. Jorba, À.: A methodology for the numerical computation of normal forms, centre manifolds and first integrals of Hamiltonian systems. Exp. Math. 8(2), 155–195 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  9. Jorba, À., Zou, M.: A software package for the numerical integration of ODE’s by means of high-order Taylor methods. Exp. Math. 14(1), 99–117 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  10. Lekien, F., Leonard, N.: Dynamically consistent Lagrangian coherent structures. Exp. Chaos 742, 132–139 (2004)Google Scholar
  11. Makino, K., Berz, M.: COSY INFINITY 9.1 Programmer’s Manual. MSU Report MSUHEP101214. Michigan State University, East Lansing, MI (2011)Google Scholar
  12. McKenzie, R., Szebehely, V.: Nonlinear stability motion around the triangular libration points. Celest. Mech. 23(3), 223–229 (1981)MathSciNetCrossRefADSGoogle Scholar
  13. PARI/GP (2012) Version 2.5.3, Bordeaux.
  14. Peng, J., Dabiri, J.O.: Transport of inertial particles by Lagrangian coherent structures: application to predator–prey interaction in jellyfish feeding. J. Fluid Mech. 623, 75–84 (2009). doi: 10.1017/S0022112008005089 zbMATHCrossRefADSGoogle Scholar
  15. Pérez, D., Gómez, G., Masdemont, J.J.: Detecting invariant manifolds using hyperbolic Lagrangian coherent structures. Paper number IAA-AAS-DyCoSS1-08-06, IAA Conference on Dynamics and Control of Space Systems. “Final papers”, Porto (2012)Google Scholar
  16. Shadden, S., Lekien, F., Marsden, J.E.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys. D Nonlinear Phenom. 212(3), 271–304 (2005)zbMATHMathSciNetCrossRefADSGoogle Scholar
  17. Short, C., Howell, K.: Lagrangian coherent structures in various map representations for application to multi-body gravitational regimes. Acta Astronaut. 94(2), 592–607 (2014)CrossRefADSGoogle Scholar
  18. Simó, C.: Global dynamics and fast indicators. Global analysis of dynamical systems 373-389, Inst. Phys., Bristol (2001)Google Scholar
  19. Simó, C., Sousa-Silva, P., Terra, M.: Practical stability domains near L4,5 in the restricted three-body problem: some preliminary facts. progress and challenges in dynamical systems. Springer Proc. Math. Stat. 54, 367–382 (2013)Google Scholar
  20. Szebehely, V.: Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, London (1967)Google Scholar
  21. Valli, M., Armellin, R., Di Lizia, P., Lavagna, M.: Nonlinear filtering methods for spacecraft navigation based on differential algebra. Acta Astronaut. 94(1), 363–374 (2014)CrossRefADSGoogle Scholar
  22. Wilczak, D.: CAPD DynSys Library Documentation. The CAPD group [online]. (2015)
  23. Wittig, A., Di Lizia, P., Armellin, R., Makino, K., Bernelli-Zazzera, F., Berz, M.: Propagation of large uncertainty sets in orbital dynamics by automatic domain splitting. Celest. Mech. Dynam. Astronom. 122, 239–261 (2015)MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Daniel Pérez-Palau
    • 1
    Email author
  • Josep J. Masdemont
    • 2
  • Gerard Gómez
    • 3
  1. 1.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain
  2. 2.IEEC and Departament de Matemàtica Aplicada I, ETSEIBUniversitat Politècnica de CatalunyaBarcelonaSpain
  3. 3.IEEC and Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

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