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The Parameterization Method for Invariant Manifolds pp 187–238Cite as

A Newton-like Method for Computing Normally Hyperbolic Invariant Tori

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Part of the book series: Applied Mathematical Sciences ((AMS,volume 195))

Abstract

This chapter presents some ideas of normally hyperbolic manifold theory, and focuses on the algorithmic application of the parameterization method in such context. The parameterization method is applied to the computation of several normally hyperbolic invariant manifolds, in the following examples: computation of an attracting invariant curve in a 2D- Fattened Arnold Family, computation of a saddle invariant curve in a 3D- Fattened Arnold Family, and the computation of a 2D normally hyperbolic invariant cylinder in the Froeschlé map.

M.C. acknowledges support from the Spanish grants MTM2009-09723, MTM2012-32541 and MTM2015-67724-P, the FPI grant BES-2010-039663, the Catalan grant 2014-SGR-1145, and the NSF grant DMS-1500943. A.H. acknowledges support from the Spanish grants MTM2009-09723, MTM2012-32541 and MTM2015-67724-P, and the Catalan grants 2009-SGR-67 and 2014-SGR-1145.

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References

  1. R. Aris, I. G. Kevrekidis, S. Pelikan, and L. D. Schmidt, Numerical computation of invariant circles of maps, Phys. D 16 (1985), no. 2, 243–251.

    Article  MathSciNet  MATH  Google Scholar 

  2. V. I. Arnold, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR 156 (1964), 9–12.

    MathSciNet  Google Scholar 

  3. V. S. Afraĭmovich and L. P. Shil′nikov, Invariant two-dimensional tori, their breakdown and stochasticity, Methods of the qualitative theory of differential equations, Gor′kov. Gos. Univ., Gorki, 1983, pp. 3–26, 164.

    Google Scholar 

  4. H. W. Broer, A. Hagen, and G. Vegter, Numerical continuation of normally hyperbolic invariant manifolds, Nonlinearity 20 (2007), no. 6, 1499–1534.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. W. Bates, K. Lu, and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc. 135 (1998), no. 645, viii+129.

    Google Scholar 

  6. H. W. Broer, H. M. Osinga, and G. Vegter, Algorithms for computing normally hyperbolic invariant manifolds, Z. Angew. Math. Phys. 48 (1997), no. 3, 480–524.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. W. Broer, C. Simó, and J.-C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity 11 (1998), 667–770.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. J. Capiński, Covering relations and the existence of topologically normally hyperbolic invariant sets, Discrete Contin. Dyn. Syst. 23 (2009), no. 3, 705–725.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Calleja, A. Celletti, and R. de la Llave, A KAM theory for conformally symplectic systems: efficient algorithms and their validation, J. Differential Equations 255 (2013), no. 5, 978–1049.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Calleja and J.-Ll. Figueras, Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map, Chaos 22 (2012), 033114.

    Google Scholar 

  11. X. Cabré, E. 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), no. 2, 283–328.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. J. Capiński, The parameterization method for invariant manifolds. II. Regularity with respect to parameters, Indiana Univ. Math. J. 52 (2003), no. 2, 329–360.

    Article  MathSciNet  Google Scholar 

  13. M. J. Capiński, The parameterization method for invariant manifolds. III. Overview and applications, J. Differential Equations 218 (2005), no. 2, 444–515.

    Article  MathSciNet  Google Scholar 

  14. L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), no. 1, 144.

    Google Scholar 

  15. M. Canadell and A. Haro, Parameterization method for computing quasi-periodic reducible normally hyperbolic invariant tori, F. Casas, V. Martínez (eds.), Advances in Differential Equations and Applications, SEMA SIMAI Springer Series, vol. 4, Springer, 2014.

    Google Scholar 

  16. M. J. Capiński, A KAM-like theorem for quasi-periodic normally hyperbolic invariant tori, Preprint, 2015.

    Google Scholar 

  17. M. J. Capiński, Parameterization methods for computing quasi-periodic normally hyperbolic invariant tori: algorithms and numerical explorations, In progress, 2015.

    Google Scholar 

  18. T. N. Chan, Numerical bifurcation analysis of simple dynamical systems, Ph.D. thesis, Department of Computer Science, Concordia University, 1983.

    Google Scholar 

  19. E. Castellà and À. Jorba, On the vertical families of two-dimensional tori near the triangular points of the bicircular problem, Celestial Mech. Dynam. Astronom. 76 (2000), no. 1, 35–54.

    Article  MathSciNet  MATH  Google Scholar 

  20. M.J. Capiński and C. Simó, Computer assisted proof for normally hyperbolic invariant manifolds, Nonlinearity 25 (2012), 1997–2026.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. W. Cooley and J. W. Tukey, An algorithm for the machine calculation of complex Fourier series, Mathematics of Computation 19 (1965), no. 90, 297–301.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. J. Capinski and P. Zgliczynski, Transition tori in the planar restricted elliptic three-body problem, Nonlinearity 24 (2011), 1395–1432.

    Article  MathSciNet  MATH  Google Scholar 

  23. L. Dieci and G. Bader, Solution of the systems associated with invariant tori approximation. II. Multigrid methods, SIAM J. Sci. Comput. 15 (1994), no. 6, 1375–1400.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Delshams, R. de la Llave, and T. M. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model, Mem. Amer. Math. Soc. 179 (2006), no. 844, viii+141.

    Google Scholar 

  25. A. Delshams and G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity 22 (2009), no. 8, 1997–2077.

    Article  MathSciNet  MATH  Google Scholar 

  26. C. Díez, À. Jorba, and C. Simó, A dynamical equivalent to the equilateral libration points of the real Earth-Moon system, Celestial Mech. 50 (1991), no. 1, 13–29.

    Article  MATH  Google Scholar 

  27. L. Dieci and J. Lorenz, Block M-matrices and computation of invariant tori, SIAM J. Sci. Statist. Comput. 13 (1992), no. 4, 885–903.

    Article  MathSciNet  MATH  Google Scholar 

  28. S. P. Diliberto, Computation of invariant tori by the method of characteristics, SIAM J. Numer. Anal. 32 (1995), no. 5, 1436–1474.

    Article  MathSciNet  Google Scholar 

  29. R. de la Llave, A. González, À. Jorba, and J. Villanueva, KAM theory without action-angle variables, Nonlinearity 18 (2005), no. 2, 855–895.

    Article  MathSciNet  MATH  Google Scholar 

  30. R. de la Llave and A. Luque, Differentiability at the tip of Arnold tongues for Diophantine rotations: numerical studies and renormalization group explanations, J. Stat. Phys. 143 (2011), no. 6, 1154–1188.

    Article  MathSciNet  MATH  Google Scholar 

  31. L. Dieci, J. Lorenz, and R. D. Russell, Numerical calculation of invariant tori, SIAM J. Sci. Statist. Comput. 12 (1991), no. 3, 607–647.

    Article  MathSciNet  MATH  Google Scholar 

  32. L. J. Díaz, I. L. Rios, and M. Viana, The intermittency route to chaotic dynamics, Global analysis of dynamical systems, Inst. Phys., Bristol, 2001, pp. 309–327.

    MATH  Google Scholar 

  33. J. Eldering, Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry, C. R. Math. Acad. Sci. Paris 350 (2012), no. 11–12, 617–620.

    Article  MathSciNet  MATH  Google Scholar 

  34. K. D. Edoh, R. D. Russell, and W. Sun, Computation of invariant tori by orthogonal collocation, Appl. Numer. Math. 32 (2000), no. 3, 273–289.

    Article  MathSciNet  MATH  Google Scholar 

  35. N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/1972), 193–226.

    Google Scholar 

  36. M. Frigo and S. G. Johnson, The design and implementation of FFTW3, Proceedings of the IEEE 93 (2005), no. 2, 216–231, Special issue on “Program Generation, Optimization, and Platform Adaptation”.

    Google Scholar 

  37. C. Froesché, Numerical study of a four-dimensional mapping, Astron. Astrophys. 16 (1972), 172–189.

    MathSciNet  Google Scholar 

  38. C. L. Fefferman and L. A. Seco, Singularity theory for non-twist KAM tori, Mem. Amer. Math. Soc. 227 (2014), no. 1067, vi+115.

    Google Scholar 

  39. T. Ge and A. Y. T. Leung, Construction of invariant torus using Toeplitz Jacobian matrices/fast Fourier transform approach, Nonlinear Dynam. 15 (1998), no. 3, 283–305.

    Article  MathSciNet  MATH  Google Scholar 

  40. S. V. Gonchenko, C. Simó, and A. Vieiro, Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight, Nonlinearity 26 (2013), no. 3, 621–678.

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Haro and R. de la Llave, Persistence of normally hyperbolic invariant manifolds, In progress.

    Google Scholar 

  42. J. K. Hale, Manifolds on the verge of a hyperbolicity breakdown, Chaos 16 (2006), 013120.

    Article  MathSciNet  MATH  Google Scholar 

  43. J. K. Hale, 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), no. 6, 1261–1300.

    Article  MathSciNet  Google Scholar 

  44. J. K. Hale, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results, J. Differential Equations 228 (2006), no. 2, 530–579.

    Article  MathSciNet  MATH  Google Scholar 

  45. J. K. Hale, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst. 6 (2007), no. 1, 142–207 (electronic).

    Google Scholar 

  46. G. Huguet, R. de la Llave, and Y. Sire, Computation of whiskered invariant tori and their associated manifolds: new fast algorithms, Discrete Contin. Dyn. Syst. 32 (2012), no. 4, 1309–1353.

    MathSciNet  MATH  Google Scholar 

  47. M. E. Henderson, Computing invariant manifolds by integrating fat trajectories, SIAM J. Appl. Dyn. Syst. 4 (2005), no. 4, 832–882 (electronic).

    Google Scholar 

  48. M. E. Henderson, Flow box tiling methods for compact invariant manifolds, SIAM J. Appl. Dyn. Syst. 10 (2011), no. 3, 1154–1176.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Huang, T. Küpper, and N. Masbaum, Computation of invariant tori by the Fourier methods, SIAM J. Sci. Comput. 18 (1997), no. 3, 918–942.

    Article  MathSciNet  MATH  Google Scholar 

  50. M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin, 1977.

    MATH  Google Scholar 

  51. À. Jorba and M. Ollé, Invariant curves near Hamiltonian-Hopf bifurcations of four-dimensional symplectic maps, Nonlinearity 17 (2004), no. 2, 691–710.

    Article  MathSciNet  MATH  Google Scholar 

  52. K. Kaneko and R. Bagley, Arnold diffusion, ergodicity and intermittency in a coupled standard mapping, Physics Letters A 110 (1985), no. 9, 435–440.

    Article  MathSciNet  Google Scholar 

  53. B. Krauskopf, H. M. Osinga, and J. Galán-Vioque (eds.), Numerical continuation methods for dynamical systems, Understanding Complex Systems, Springer, Dordrecht, 2007, Path following and boundary value problems, Dedicated to Eusebius J. Doedel for his 60th birthday.

    Google Scholar 

  54. J. Lorenz and A. Morlet, Numerical solution of a functional equation on a circle, SIAM J. Numer. Anal. 29 (1992), no. 6, 1741–1768.

    Article  MathSciNet  MATH  Google Scholar 

  55. R. Mañé, Persistent manifolds are normally hyperbolic, Trans. Amer. Math. Soc. 246 (1978), 261–283.

    Article  MathSciNet  MATH  Google Scholar 

  56. J. N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math. 30 (1968), 479–483.

    Google Scholar 

  57. J. M. Mondelo, E. Barrabés, G. Gómez, and M. Ollé, Numerical parametrisations of libration point trajectories and their invariant manifolds, AAS/AIAA Astrodynamics Specialists Conference, AAS, 2007.

    Google Scholar 

  58. J. M. Mondelo, Fast numerical computation of Lissajous and quasi-halo libration point trajectories and their invariant manifolds, Paper IAC-12, C1, 6, 9, x14982. 63rd International Astronautical Congress, Naples, Italy, 2012.

    Google Scholar 

  59. R. S. MacKay, J. D. Meiss, and J. Stark, Converse KAM theory for symplectic twist maps, Nonlinearity 2 (1989), no. 4, 555–570.

    Article  MathSciNet  MATH  Google Scholar 

  60. M. J. Mohlenkamp, A fast transform for spherical harmonics, J. Fourier Anal. Appl. 5 (1999), no. 2–3, 159–184.

    Article  MathSciNet  MATH  Google Scholar 

  61. G. Moore, Computation and parametrization of periodic and connecting orbits, IMA J. Numer. Anal. 15 (1995), no. 2, 245–263.

    Article  MathSciNet  MATH  Google Scholar 

  62. R. E. Moore, Computation and parameterisation of invariant curves and tori, SIAM J. Numer. Anal. 33 (1996), no. 6, 2333–2358.

    Article  MathSciNet  MATH  Google Scholar 

  63. B. B. Peckham and F. Schilder, Computing Arnol′d tongue scenarios, J. Comput. Phys. 220 (2007), no. 2, 932–951.

    Article  MathSciNet  MATH  Google Scholar 

  64. B. Rasmussen and L. Dieci, A geometrical method for the approximation of invariant tori, J. Comput. Appl. Math. 216 (2008), no. 2, 388–412.

    Article  MathSciNet  MATH  Google Scholar 

  65. V. Reichelt, Computing invariant tori and circles in dynamical systems, Numerical methods for bifurcation problems and large-scale dynamical systems (Minneapolis, MN, 1997), IMA Vol. Math. Appl., vol. 119, Springer, New York, 2000, pp. 407–437.

    Google Scholar 

  66. C. Simó, On the Analytical and Numerical Approximation of Invariant Manifolds, Modern Methods in Celestial Mechanics, Comptes Rendus de la 13ieme Ecole Printemps d’Astrophysique de Goutelas (France), 24–29 Avril, 1989. Edited by Daniel Benest and Claude Froeschlé. Gif-sur-Yvette: Editions Frontieres, 1990., p.285 (1990), 285–330.

    Google Scholar 

  67. J. Sánchez, M. Net, and C. Simó, Computation of invariant tori by Newton-Krylov methods in large-scale dissipative systems, Phys. D 239 (2010), no. 3–4, 123–133.

    Article  MathSciNet  MATH  Google Scholar 

  68. F. Schilder, H. M. Osinga, and W. Vogt, Continuation of quasi-periodic invariant tori, SIAM J. Appl. Dyn. Syst. 4 (2005), no. 3, 459–488 (electronic).

    Google Scholar 

  69. F. Schilder, W. Vogt, S. Schreiber, and H. M. Osinga, Fourier methods for quasi-periodic oscillations, Internat. J. Numer. Methods Engrg. 67 (2006), no. 5, 629–671.

    Article  MathSciNet  MATH  Google Scholar 

  70. R. Swanson, The spectral characterization of normal hyperbolicity, Proc. Amer. Math. Soc. 89 (1983), no. 3, 503–509.

    Article  MathSciNet  MATH  Google Scholar 

  71. M. R. Trummer, Spectral methods in computing invariant tori, Appl. Numer. Math. 34 (2000), no. 2–3, 275–292, Auckland numerical ordinary differential equations (Auckland, 1998).

    Google Scholar 

  72. M. van Veldhuizen, A new algorithm for the numerical approximation of an invariant curve, SIAM J. Sci. Stat. Comput. 8 (1987), no. 6, 951–962.

    Article  MathSciNet  MATH  Google Scholar 

  73. S. Wiggins, Normally hyperbolic invariant manifolds in dynamical systems, Applied Mathematical Sciences, vol. 105, Springer-Verlag, New York, 1994.

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, USA

    Marta Canadell

  2. Barcelona Graduate School of Mathematics, Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Spain

    Àlex Haro

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  1. Marta Canadell
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Canadell, M., Haro, À. (2016). A Newton-like Method for Computing Normally Hyperbolic Invariant Tori. In: The Parameterization Method for Invariant Manifolds. Applied Mathematical Sciences, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-29662-3_5

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