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The Parameterization Method for Quasi-Periodic Systems: From Rigorous Results to Validated Numerics

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The Parameterization Method for Invariant Manifolds

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

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Abstract

This chapter developes the “from theory-to algorithms-to computations-to validations” program for response tori in quasi-periodically forced systems. First, it provides a full proof of a Kantorovich-like theorem for invariant tori in discrete quasi-periodic systems. The proof of this theorem leads to several algorithms for the computation of invariant tori in this context, that are also detailed. Next, it is explained a computer assisted methodology for the validation of numerical results based on the previous a posteriori theorem. The chapter ends with three examples: validation of saddle invariant tori on the verge of breakdown, computation of a rigorous upper bound of the measure of Cantor-like spectra of a discrete Schrödinger operator, and validation of an attracting torus that by direct double precision seems to be a strange nonchaotic attractor.

J.-L.F. acknowledges support from the Spanish grants MTM2009-09723, MTM2012-32541 and the Catalan grant 2009-SGR-67. 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. D. V. Anosov, S. Kh. Aranson, V. I. Arnold, I. U. Bronshtein, Yu. S. Il’yashenko, and V. Z. Grines, Ordinary differential equations and smooth dynamical systems, Springer-Verlag New York, Inc., New York, NY, USA, 1997.

    Google Scholar 

  2. A. Abad, R. Barrio, F. Blesa, and M. Rodríguez, Algorithm 924: Tides, a Taylor series integrator for differential equations, ACM Trans. Math. Softw. 39 (2012), no. 1, 5:1–5:28.

    Google Scholar 

  3. E. L. Allgower and K. Georg, Introduction to numerical continuation methods, Classics in Applied Mathematics, vol. 45, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003, Reprint of the 1990 edition Springer-Verlag, Berlin;.

    Google Scholar 

  4. G. Arioli and H. Koch, The critical renormalization fixed point for commuting pairs of area-preserving maps, Comm. Math. Phys. 295 (2010), no. 2, 415–429.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. A. Adomaitis, I. G. Kevrekidis, and R. de la Llave, Predicting the complexity of disconnected basins of attraction for a noninvertible system, Tech. report, Systems Research Center Technical Report TR91-41, 1991.

    Google Scholar 

  6. R. A. Adomaitis, I. G. Kevrekidis, and R., A computer-assisted study of global dynamic transitions for a noninvertible system, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 17 (2007), no. 4, 1305–1321.

    Google Scholar 

  7. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, third ed., Encyclopaedia of Mathematical Sciences, vol. 3, Springer-Verlag, Berlin, 2006.

    Google Scholar 

  8. 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 

  9. G. Arioli, H. Koch, and S. Terracini, Two novel methods and multi-mode periodic solutions for the Fermi-Pasta-Ulam model, Comm. Math. Phys. 255 (2005), no. 1, 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Abraham and J. E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1978.

    Google Scholar 

  11. Ll. Alsedà and M. Misiurewicz, Attractors for unimodal quasiperiodically forced maps, J. Difference Equ. Appl. 14 (2008), no. 10–11, 1175–1196.

    Google Scholar 

  12. D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209.

    MathSciNet  Google Scholar 

  13. V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 13–40.

    Google Scholar 

  14. V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Russ. Math. Surveys 18 (1963), 85–192.

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  16. V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 250, Springer-Verlag, New York, 1983, Translated from the Russian by Joseph Szücs, Translation edited by Mark Levi.

    Google Scholar 

  17. V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 250, Springer-Verlag, New York, 1988, Translated from the Russian by Joseph Szücs [József M. Szűcs].

    Google Scholar 

  18. 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 

  19. AUTODIFF: Community Portal for Automatic Differentiation. Accessed March 14, 2016, http://www.autodiff.org.

  20. R. Barrio, Sensitivity analysis of ODEs/DAEs using the Taylor series method, SIAM J. Sci. Comput. 27 (2006), no. 6, 1929–1947 (electronic).

    Google Scholar 

  21. W.-J. Beyn, A. Champneys, E. Doedel, W. Govaerts, Y.A. Kuznetsov, and B. Sandstede, Numerical continuation, and computation of normal forms, Handbook of dynamical systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 149–219.

    MATH  Google Scholar 

  22. C. Bischof, G. Corliss, and A. Griewank, Structured second- and higher-order derivatives through univariate Taylor series, Optim. Methods Software 2 (1993), 211–232.

    Article  Google Scholar 

  23. M. Berz, Algorithms for higher derivatives in many variables with applications to beam physics, Automatic differentiation of algorithms (Breckenridge, CO, 1991), SIAM, Philadelphia, PA, 1991, pp. 147–156.

    Google Scholar 

  24. M. Berti, Nonlinear oscillations of Hamiltonian PDEs, Progress in Nonlinear Differential Equations and their Applications, 74, Birkhäuser Boston Inc., Boston, MA, 2007.

    Google Scholar 

  25. I. Baldomá, E. Fontich, R. de la Llave, and P. Martín, The parameterization method for one-dimensional invariant manifolds of higher dimensional parabolic fixed points, Discrete Contin. Dyn. Syst. 17 (2007), no. 4, 835–865.

    Article  MathSciNet  MATH  Google Scholar 

  26. I. Baldomá, E. Fontich, and P. Martín, Invariant manifolds of parabolic fixed points (I). existence and dependence of parameters, 2015.

    Google Scholar 

  27. W.-J. Beyn, Invariant manifolds of parabolic fixed points (II). approximations by sums of homogeneous functions, 2015.

    Google Scholar 

  28. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, A proof of Kolmogorov’s theorem on invariant tori using canonical transformations defined by the Lie method, Nuovo Cimento B (11) 79 (1984), no. 2, 201–223.

    Google Scholar 

  29. I. Baldomá and A. Haro, One dimensional invariant manifolds of Gevrey type in real-analytic maps, Discrete Contin. Dyn. Syst. Ser. B 10 (2008), no. 2–3, 295–322.

    MathSciNet  MATH  Google Scholar 

  30. H. W. Broer, H. Hanßmann, À. Jorba, J. Villanueva, and F. Wagener, Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach, Nonlinearity 16 (2003), no. 5, 1751–1791.

    Article  MathSciNet  MATH  Google Scholar 

  31. F. Blomquist, W. Hofschuster, and W. Krämer, Real and complex Taylor arithmetic in c-xsc, Preprint, 2005.

    Google Scholar 

  32. H. W. Broer, G. B. Huitema, and M. B. Sevryuk, Quasi-periodic motions in families of dynamical systems. Order amidst chaos, Lecture Notes in Math., Vol 1645, Springer-Verlag, Berlin, 1996.

    Google Scholar 

  33. H. W. Broer, G. B. Huitema, and F. Takens, Unfoldings of quasi-periodic tori, Mem. Amer. Math. Soc. 83 (1990), no. 421, 1–81, 171–175.

    Google Scholar 

  34. 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 

  35. F. Biscani, Design and implementation of a modern algebraic manipulator for Celestial Mechanics, Ph.D. thesis, Dipartimento di Astronomia, Università di Padova, 2008.

    Google Scholar 

  36. K. Bjerklöv, SNA’s in the quasi-periodic quadratic family, Comm. Math. Phys. 286 (2009), no. 1, 137–161.

    Article  MathSciNet  MATH  Google Scholar 

  37. R. P. Brent and H. T. Kung, Fast algorithms for manipulating formal power series, J. Assoc. Comput. Mach. 25 (1978), no. 4, 581–595.

    Article  MathSciNet  MATH  Google Scholar 

  38. W.-J. Beyn and W. Kleß, Numerical Taylor expansions of invariant manifolds in large dynamical systems, Numer. Math. 80 (1998), no. 1, 1–38.

    Article  MathSciNet  MATH  Google Scholar 

  39. 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 

  40. N. N. Bogoliubov and Yu. A. Mitropolsky, The method of integral manifolds in nonlinear mechanics, Contributions to Differential Equations 2 (1963), 123–196 (1963).

    MathSciNet  Google Scholar 

  41. E. M. Bollt and J. D. Meiss, Breakup of invariant tori for the four-dimensional semi-standard map, Phys. D 66 (1993), no. 3–4, 282–297.

    Article  MathSciNet  MATH  Google Scholar 

  42. M. Bodrato, Towards optimal Toom-Cook multiplication for univariate and multivariate polynomials in characteristic 2 and 0, WAIFI 2007 proceedings (C. Carlet and B. Sunar, eds.), LNCS, vol. 4547, Springer, June 2007, pp. 116–133.

    Google Scholar 

  43. J. B. Bost, Tores invariants des systèmes dynamiques hamiltoniens (d’après Kolmogorov, Arnold, Moser, Rüssmann, Zehnder, Herman, Pöschel, …), Astérisque (1986), no. 133–134, 113–157, Seminar Bourbaki, Vol. 1984/85.

    Google Scholar 

  44. J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Internat. Math. Res. Notices (1994), no. 11, 475ff., approx. 21 pp. (electronic).

    Google Scholar 

  45. J. B. Bost, On Melnikov’s persistency problem, Math. Res. Lett. 4 (1997), no. 4, 445–458.

    Google Scholar 

  46. J. B. Bost, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math. 2 (1998), no. 148, 363–439.

    Google Scholar 

  47. 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 

  48. H. W. Broer, KAM theory: the legacy of A. N. Kolmogorov’s 1954 paper. Comment on: “The general theory of dynamic systems and classical mechanics” (French) [in proceedings of the international congress of mathematicians, amsterdam, 1954, vol. 1, 315–333, Erven P. Noordhoff N.V., Groningen, 1957], Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 4, 507–521.

    Google Scholar 

  49. M. Brin and G. Stuck, Introduction to dynamical systems, Cambridge University Press, Cambridge, 2002.

    Book  MATH  Google Scholar 

  50. K. Bjerklöv and M. Saprykina, Universal asymptotics in hyperbolicity breakdown, Nonlinearity 21 (2008), no. 3, 557–586.

    Article  MathSciNet  MATH  Google Scholar 

  51. 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 

  52. H. W. Broer, C. Simó, and R. Vitolo, Chaos and quasi-periodicity in diffeomorphisms of the solid torus, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 3, 871–905.

    Article  MathSciNet  MATH  Google Scholar 

  53. M. Canadell, Computation of normally hyperbolic invariant manifolds, Ph.D. thesis, Departament de Matemàtica Aplicada i Analísi, Universitat de Barcelona, 2014.

    Google Scholar 

  54. CAPD: Computer Assisted Proofs in Dynamics group. Accessed March 14, 2016, http://capd.ii.uj.edu.pl/download.php.

  55. 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 

  56. A. Celletti and L. Chierchia, Construction of Analytic KAM Surfaces and Effective Stability Bounds, Comm. Math. Phys. 118 (1988), no. 1, 199–161.

    Article  MathSciNet  MATH  Google Scholar 

  57. M. J. Capiński, On the stability of realistic three-body problems, Comm. Math. Phys. 186 (1997), no. 2, 413–449.

    Article  MathSciNet  Google Scholar 

  58. M. J. Capiński, KAM stability and celestial mechanics, Mem. Amer. Math. Soc. 187 (2007), no. 878, viii+134.

    Google Scholar 

  59. 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 

  60. S. Coffey, A. Deprit, E. Deprit, L. Healy, and B. R. Miller, A toolbox for nonlinear dynamics, Computer aided proofs in analysis. (Cincinnati, OH, 1989) (Kenneth R. Meyer and Dieter S. Schmidt, eds.), IMA Vol. Math. Appl., vol. 28, Springer-Verlag, New York, 1991, pp. 97–115.

    Google Scholar 

  61. R. Calleja and R. de la Llave, Fast numerical computation of quasi-periodic equilibrium states in 1D statistical mechanics, including twist maps, Nonlinearity 22 (2009), no. 6, 1311–1336.

    Article  MathSciNet  MATH  Google Scholar 

  62. M. J. Capiński, A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification, Nonlinearity 23 (2010), no. 9, 2029–2058.

    Article  MathSciNet  MATH  Google Scholar 

  63. A. Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001.

    Book  MATH  Google Scholar 

  64. X. Cabré and E. Fontich, Regularity and uniqueness of one dimensional invariant manifolds. Accessed March 14, 2016, http://www.maia.ub.es/dsg/1994/9401Cabre.pdf, 1994.

  65. 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 

  66. 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 

  67. 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 

  68. 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 

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

    Google Scholar 

  70. 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 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  74. B. V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52 (1979), no. 5, 264–379.

    Article  MathSciNet  Google Scholar 

  75. C. Chicone, Ordinary differential equations with applications, second ed., Texts in Applied Mathematics, vol. 34, Springer, New York, 2006.

    Google Scholar 

  76. CHOMP: Computational Homology Project. Accessed March 14, 2016, http://chomp.rutgers.edu/index.html.

  77. A. Chenciner and G. Iooss, Bifurcations de tores invariants, Arch. Rational Mech. Anal. 69 (1979), no. 2, 109–198.

    Article  MathSciNet  MATH  Google Scholar 

  78. B. V. Chirikov, Persistance et bifurcation de tores invariants, Arch. Rational Mech. Anal. 71 (1979), no. 4, 301–306.

    MathSciNet  MATH  Google Scholar 

  79. 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 

  80. M.-C. Ciocci, A. Litvak-Hinenzon, and H. Broer, Survey on dissipative KAM theory including quasi-periodic bifurcation theory, Geometric mechanics and symmetry, London Math. Soc. Lecture Note Ser., vol. 306, Cambridge Univ. Press, Cambridge, 2005, pp. 303–355.

    Google Scholar 

  81. R. Castelli, J.-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), no. 1, 132–167.

    Article  MathSciNet  MATH  Google Scholar 

  82. E. Canalias and J. Masdemont, Computing natural transfers between Sun-Earth and Earth-Moon Lissajous libration point orbits, Acta Astronautica 63 (2008), no. 1–4, 238–248.

    Article  Google Scholar 

  83. Coin3D: 3D Graphics Toolkit. Accessed March 14, 2016, http://www.coin3d.org/.

  84. C. C. Conley, Low energy transit orbits in the restricted three-body problem, SIAM J. Appl. Math. 16 (1968), 732–746.

    Article  MathSciNet  MATH  Google Scholar 

  85. S. A. Cook, On the minimum computation time of functions, Ph.D. thesis, Department of Mathematics, Harvard University, 1966.

    Google Scholar 

  86. COSY INFINITY. Accessed March 14, 2016, http://www.bt.pa.msu.edu/index_cosy.htm.

  87. C. Q. Cheng and Y. S. Sun, Existence of KAM tori in degenerate systems, J. Differential equations 114 (1994), no. 1, 288–335.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  89. 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 

  90. C-XSC. Accessed March 14, 2016, http://www2.math.uni-wuppertal.de/~xsc/xsc/cxsc.html.

  91. 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 

  92. D. Damanik, The spectrum of the Almost-Mathieu operator, Preprint, 2009.

    MATH  Google Scholar 

  93. 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 

  94. W.-H. Du and W.-J. Beyn, The numerical approximation of center manifolds in Hamiltonian systems, J. Math. Anal. Appl. 288 (2003), no. 1, 28–46.

    Article  MathSciNet  MATH  Google Scholar 

  95. A. Delshams and R. de la Llave, KAM theory and a partial justification of Greene’s criterion for nontwist maps, SIAM J. Math. Anal. 31 (2000), no. 6, 1235–1269 (electronic).

    Google Scholar 

  96. 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 

  97. M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math. 75 (1997), no. 3, 293–317.

    Article  MathSciNet  MATH  Google Scholar 

  98. 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 

  99. S. P. Diliberto, Perturbation theorems for periodic surfaces. I. Definitions and main theorems, Rend. Circ. Mat. Palermo (2) 9 (1960), 265–299.

    Google Scholar 

  100. 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 

  101. E. Doedel, H. B. Keller, and J.-P. Kernévez, Numerical analysis and control of bifurcation problems. I. Bifurcation in finite dimensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1 (1991), no. 3, 493–520.

    Article  MathSciNet  MATH  Google Scholar 

  102. E. Doedel, B. Krauskopf, and H. M. Osinga, Global bifurcations of the Lorenz manifold, Nonlinearity 19 (2006), no. 12, 2947–2972.

    Article  MathSciNet  MATH  Google Scholar 

  103. 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 

  104. 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 

  105. R. de la Llave, Hyperbolic dynamical systems and generation of magnetic fields by perfectly conducting fluids, Geophys. Astrophys. Fluid Dynam. 73 (1993), no. 1–4, 123–131, Magnetohydrodynamic stability and dynamos (Chicago, IL, 1992).

    Google Scholar 

  106. S. P. Diliberto, Invariant manifolds associated to nonresonant spectral subspaces, J. Statist. Phys. 87 (1997), no. 1–2, 211–249.

    MathSciNet  MATH  Google Scholar 

  107. S. P. Diliberto, A tutorial on KAM theory, Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 175–292.

    Google Scholar 

  108. 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 

  109. 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 

  110. R. de la Llave and J. D. Mireles James, Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps, Discrete Contin. Dyn. Syst. 32 (2012), no. 12, 4321–4360.

    Article  MathSciNet  MATH  Google Scholar 

  111. R. de la Llave and A. Olvera, The obstruction criterion for non-existence of invarian circles and renormalization, Nonlinearity 19 (2006), no. 8, 1907–1937.

    Article  MathSciNet  MATH  Google Scholar 

  112. R. de la Llave and D. Rana, Accurate strategies for small divisor problems, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 85–90.

    Google Scholar 

  113. S. P. Diliberto, Accurate strategies for K.A.M. bounds and their implementation, Computer aided proofs in analysis (Cincinnati, OH, 1989), IMA Vol. Math. Appl., vol. 28, Springer, New York, 1991, pp. 127–146.

    Google Scholar 

  114. R. de la Llave and C. E. Wayne, Whiskered and low dimensional tori in nearly integrable Hamiltonian systems, Math. Phys. Electron. J. 10 (2004), Paper 5, 45 pp. (electronic).

    Google Scholar 

  115. 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 

  116. S. Datta, R. Ramaswamy, and A. Prasad, Fractalization route to strange nonchaotic dynamics, Phys. Rev. E 70 (2004), no. 4, 046203.

    Google Scholar 

  117. A. Delshams and R. Ramírez-Ros, Singular separatrix splitting and the Melnikov method: an experimental study, Experiment. Math. 8 (1999), no. 1, 29–48.

    Article  MathSciNet  MATH  Google Scholar 

  118. 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 

  119. H. S. Dumas, The KAM story, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014, A friendly introduction to the content, history, and significance of classical Kolmogorov-Arnold-Moser theory.

    Google Scholar 

  120. 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 

  121. L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 1, 115–147 (1989).

    Google Scholar 

  122. L. H. Eliasson, Biasymptotic solutions of perturbed integrable Hamiltonian systems, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 1, 57–76.

    Google Scholar 

  123. L. H. Eliasson, Almost reducibility of linear quasi-periodic systems, Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 679–705.

    Google Scholar 

  124. 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 

  125. E. Fontich, R. de la Llave, and Y. Sire, Construction of invariant whiskered tori by a parameterization method. I. Maps and flows in finite dimensions, J. Differential Equations 246 (2009), no. 8, 3136–3213.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  127. J. E. Fornæss and E. A. Gavosto, Existence of generic homoclinic tangencies for Hénon mappings, J. Geom. Anal. 2 (1992), no. 5, 429–444.

    Article  MathSciNet  MATH  Google Scholar 

  128. J.-Ll. Figueras and A. Haro, Reliable computation of robust response tori on the verge of breakdown, SIAM J. Appl. Dyn. Syst. 11 (2012), 597–628.

    Google Scholar 

  129. J.-Ll. Figueras and, Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps, Chaos 25 (2015), 123119.

    Google Scholar 

  130. J.-Ll. Figueras, A. Haro, and A. Luque, Rigorous computer assisted application of KAM theory: a modern approach. Preprint available at arXiv:1601.00084.

    Google Scholar 

  131. J.-Ll. Figueras, Fiberwise Hyperbolic Invariant Tori in quasiperiodically skew product systems, Ph.D. thesis, Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, 2011.

    Google Scholar 

  132. 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 

  133. A. Farrés and À. Jorba, On the high order approximation of the centre manifold for odes, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 3, 977–1000.

    Article  MathSciNet  MATH  Google Scholar 

  134. A. M. Fox and J. D. Meiss, Critical invariant circles in asymmetric and multiharmonic generalized standard maps, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), no. 4, 1004–1026.

    Article  MathSciNet  Google Scholar 

  135. V. Franceschini and L. Russo, Stable and unstable manifolds of the Hénon mapping, J. Statist. Phys. 25 (1981), no. 4, 757–769.

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  137. E. Fontich and C. Simó, The Splitting of Separatrices for Analytic Diffeomorphism, Ergod. Th. and Dynam. Sys. 10 (1990), 295–318.

    MATH  Google Scholar 

  138. C. L. Fefferman and L. A. Seco, Interval arithmetic in quantum mechanics, Applications of interval computations (El Paso, TX, 1995), Appl. Optim., vol. 3, Kluwer Acad. Publ., Dordrecht, 1996, pp. 145–167.

    Google Scholar 

  139. G. Gallavotti, Perturbation theory for classical Hamiltonian systems, Scaling and self-similarity in physics (Bures-sur-Yvette, 1981/1982), Progr. Phys., vol. 7, Birkhäuser Boston, Boston, MA, 1983, pp. 359–426.

    Google Scholar 

  140. G. Giorgilli and L. Galgani, Formal integrals for an autonomous hamiltonian system near an equilibrium point, Cel. Mech. 17 (1978), 267–280.

    Article  MathSciNet  MATH  Google Scholar 

  141. G. Gallavotti and G. Gentile, Hyperbolic low-dimensional invariant tori and summations of divergent series, Comm. Math. Phys. 227 (2002), no. 3, 421–460.

    Article  MathSciNet  MATH  Google Scholar 

  142. J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1990, Revised and corrected reprint of the 1983 original.

    Google Scholar 

  143. A. González, A. Haro, and R. de la Llave, An introduction to singularity theory for non-twist KAM tori, In progress.

    Google Scholar 

  144. 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 

  145. A. Giorgilli, A computer program for integrals of motion, Comp. Phys. Comm. 16 (1979), 331–343.

    Article  Google Scholar 

  146. F. Gabern, À. Jorba, and U. Locatelli, On the construction of the Kolmogorov normal form for the Trojan asteroids, Nonlinearity 18 (2005), no. 4, 1705–1734.

    Article  MathSciNet  MATH  Google Scholar 

  147. G. Gómez, À. Jorba, J. Masdemont, and C. Simó, Study refinement of semi-analytical halo orbit theory, Tech. report, European Space Agency, 1991.

    Google Scholar 

  148. G. Gómez, À. Jorba, C. Simó, and J. Masdemont, Dynamics and mission design near libration point orbits - volume III: Advanced methods for collinear points, World Scientific Monograph Series in Mathematics, vol. 4, World Scientific Publishing Co. Inc., River Edge, NJ, 2001. Reprint of ESA Report Study Refinement of Semi-Analytical Halo Orbit Theory, 1991.

    Google Scholar 

  149. C. L. Fefferman and L. A. Seco, Dynamics and mission design near libration point orbits - volume IV: Advanced methods for triangular points, World Scientific Monograph Series in Mathematics, vol. 5, World Scientific Publishing Co. Inc., River Edge, NJ, 2001. Reprint of ESA Report Study of Poincaré Maps for Orbits Near Lagrangian Points, 1993.

    Google Scholar 

  150. A. Griewank, D. Juedes, and J. Utke, Algorithm 755: ADOL-C: A package for the automatic differentiation of algorithms written in C/C++, ACM Transactions on Mathematical Software 22 (1996), no. 2, 131–167.

    Article  MATH  Google Scholar 

  151. G. Gómez, W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont, and S. D. Ross, Connecting orbits and invariant manifolds in the spatial restricted three-body problem, Nonlinearity 17 (2004), no. 5, 1571–1606.

    Article  MathSciNet  MATH  Google Scholar 

  152. F. Gabern, W. S. Koon, J. E. Marsden, and S. D. Ross, Theory and computation of non-RRKM lifetime distributions and rates in chemical systems with three or more degrees of freedom, Phys. D 211 (2005), no. 3–4, 391–406.

    Article  MathSciNet  MATH  Google Scholar 

  153. 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 

  154. M. Gastineau and J. Laskar, Development of trip: Fast sparse multivariate polynomial multiplication using burst tries, International Conference on Computational Science (2), 2006, pp. 446–453.

    Google Scholar 

  155. G. Gómez, J. Llibre, R. Martínez, and C. Simó, Dynamics and mission design near libration points. Vol. I, World Scientific Monograph Series in Mathematics, vol. 2, World Scientific Publishing Co. Inc., River Edge, NJ, 2001, Fundamentals: the case of collinear libration points, With a foreword by Walter Flury.

    Google Scholar 

  156. J. Guckenheimer and B. Meloon, Computing periodic orbits and their bifurcations with automatic differentiation, SIAM J. Sci. Comput. 22 (2000), no. 3, 951–985 (electronic).

    Google Scholar 

  157. G. Gómez and J. M. Mondelo, The dynamics around the collinear equilibrium points of the RTBP, Phys. D 157 (2001), no. 4, 283–321.

    Article  MathSciNet  MATH  Google Scholar 

  158. G. Gómez, J. M. Mondelo, and C. Simó, A collocation method for the numerical Fourier analysis of quasi-periodic functions. I. Numerical tests and examples, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 1, 41–74.

    Article  MathSciNet  MATH  Google Scholar 

  159. C. Grebogi, E. Ott, S. Pelikan, and J. A. Yorke, Strange attractors that are not chaotic, Phys. D 13 (1984), no. 1–2, 261–268.

    Article  MathSciNet  MATH  Google Scholar 

  160. P. Gomis-Porqueras and A. Haro, A geometric description of a macroeconomic model with a center manifold, J. Econom. Dynam. Control 33 (2009), no. 6, 1217–1235.

    Article  MathSciNet  MATH  Google Scholar 

  161. S. M. Graff, On the conservation of hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations 15 (1974), 1–69.

    Article  MathSciNet  MATH  Google Scholar 

  162. J. M. Greene, A method for determining a stochastic transition, J. Math. Phys 20 (1975), no. 6, 1183–1201.

    Article  Google Scholar 

  163. A. Griewank, Evaluating derivatives, Frontiers in Applied Mathematics, vol. 19, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000, Principles and techniques of algorithmic differentiation.

    Google Scholar 

  164. V. Gelfreich and C. Simó, High-precision computations of divergent asymptotic series and homoclinic phenomena, Discrete Contin. Dyn. Syst. Ser. B 10 (2008), no. 2–3, 511–536.

    MathSciNet  MATH  Google Scholar 

  165. V. Gelfreich, C. Simó, and A. Vieiro, Dynamics of 4d symplectic maps near a double resonance, Physica D 243 (2013), no. 1, 92–110.

    Article  MathSciNet  MATH  Google Scholar 

  166. 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 

  167. A. Griewank, J. Utke, and A. Walther, Evaluating higher derivative tensors by forward propagation of univariate Taylor series, Math. Comp. 69 (2000), no. 231, 1117–1130.

    Article  MathSciNet  MATH  Google Scholar 

  168. J. Guckenheimer and A. Vladimirsky, A fast method for approximating invariant manifolds, SIAM Journal of Applied Dynamical Systems 3 (2004), 2004.

    Article  MathSciNet  MATH  Google Scholar 

  169. A. Griewank and A. Walther, Evaluating derivatives, second ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008, Principles and techniques of algorithmic differentiation.

    Google Scholar 

  170. J. Hadamard, Sur l’itération et les solutions asymptotiques des équations différentielles., S. M. F. Bull. 29 (1901), 224–228 (French).

    Google Scholar 

  171. J. K. Hale, Integral manifolds of perturbed differential systems, Ann. of Math. (2) 73 (1961), 496–531.

    Google Scholar 

  172. H. Hanssmann, Non-degeneracy conditions in KAM theory, Indagationes Mathematicae 22 (2011), no. 3–4, 241–256.

    Article  MathSciNet  MATH  Google Scholar 

  173. A. Haro, The primitive function of an exact symplectomorphism, Ph.D. thesis, Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, 1998.

    Google Scholar 

  174. J. K. Hale, Center and center-(un)stable manifolds of elliptic-hyperbolic fixed points of 4D-symplectic maps. An example: the Froeschlé map, Hamiltonian systems with three or more degrees of freedom (S’Agaró, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 533, Kluwer Acad. Publ., Dordrecht, 1999, pp. 403–407.

    Google Scholar 

  175. J. K. Hale, The primitive function of an exact symplectomorphism, Nonlinearity 13 (2000), no. 5, 1483–1500.

    Article  MathSciNet  MATH  Google Scholar 

  176. J. K. Hale, An algorithm to generate canonical transformations: application to normal forms, Phys. D 167 (2002), no. 3–4, 197–217.

    MathSciNet  Google Scholar 

  177. J. K. Hale, Automatic differentiation tools in computational dynamical systems, 2008, Lectures in the Advanced School on Specific Algebraic Manipulators (Barcelona, September 12–15, 2007).

    Google Scholar 

  178. J. K. Hale, Automatic differentiation tools in computational dynamical systems: invariant manifolds and normal forms of vector fields at fixed points, 2011, Lectures in the IMA New directions short course Invariant Objects in Dynamical Systems and their Applications (Minnesota, June 20-July 1, 2011).

    Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  181. 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.

    Google Scholar 

  182. 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 

  183. 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 

  184. 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), no. 4, 1763–1802.

    Article  MathSciNet  MATH  Google Scholar 

  185. 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 

  186. M. E. Henderson, Multiple parameter continuation: computing implicitly defined k-manifolds, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 12 (2002), no. 3, 451–476.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  188. 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 

  189. M.-R. Herman, Sur les courbes invariantes par les difféomorphismes de l’anneau. Vol. 2, Astérisque (1986), no. 144, 248, With a correction to: On the curves invariant under diffeomorphisms of the annulus, Vol. 1 (French) [Astérisque No. 103–104, Soc. Math. France, Paris, 1983].

    Google Scholar 

  190. M.-R. Herman, Inégalités “a priori” pour des tores lagrangiens invariants par des difféomorphismes symplectiques, Inst. Hautes Études Sci. Publ. Math. (1989), no. 70, 47–101 (1990).

    Google Scholar 

  191. M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astronom. J. 69 (1964), 73–79.

    Article  MathSciNet  Google Scholar 

  192. J. F. Heagy and S. M. Hammel, The birth of strange nonchaotic attractors, Phys. D 70 (1994), no. 1–2, 140–153.

    Article  MathSciNet  MATH  Google Scholar 

  193. W. Hofschuster and W. Kraemer, A Fast Public Domain Interval Library in ANSI C, Proceedings of the 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics 2 (1997), 395–400.

    Google Scholar 

  194. B. Hasselblatt and A. Katok, A first course in dynamics, Cambridge University Press, New York, 2003, With a panorama of recent developments.

    Google Scholar 

  195. 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 

  196. Y. Han, Y. Li, and Y. Yi, Degenerate lower-dimensional tori in Hamiltonian systems, J. Differential Equations 227 (2006), no. 2, 670–691.

    Article  MathSciNet  MATH  Google Scholar 

  197. J. F. Heagy and S. M. Hammel, Invariant tori in Hamiltonian systems with high order proper degeneracy, Ann. Henri Poincaré 10 (2010), no. 8, 1419–1436.

    Article  MathSciNet  Google Scholar 

  198. A. Haro, J. M. Mondelo, and B. F. Villac, Dynamical characterization of 1:1 resonance crossing trajectories at Vesta, Paper AAS 12–131, 22nd AAS/AIAA Space Flight Mechanics Meeting, January 29 - February 2, Charleston, South Carolina, USA, 2012.

    Google Scholar 

  199. A. J. Homburg, Invariant manifolds near hyperbolic fixed points, J. Difference Equ. Appl. 12 (2006), no. 10, 1057–1068.

    Article  MathSciNet  MATH  Google Scholar 

  200. M. W. Hirsch and C. C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163.

    Google Scholar 

  201. A. Haro and J. Puig, Strange nonchaotic attractors in Harper maps, Chaos 16 (2006), 033127.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  203. J. Hass and R. Schlafly, Double bubbles minimize, Ann. of Math. (2) 151 (2000), no. 2, 459–515.

    Google Scholar 

  204. A. Haro and C. Simó, To be or not to be a SNA: That is the question. Accessed March 14, 2016, http://www.maia.ub.es/dsg/2005/0503haro.pdf.

  205. G. Huguet, The role of hyperbolic invariant objects: from Arnold difussion to biological clocks, Ph.D. thesis, Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, 2008.

    Google Scholar 

  206. H.-L. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, J. Dynam. Differential Equations 20 (2008), no. 4, 831–866.

    Article  MathSciNet  MATH  Google Scholar 

  207. H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2011, Reprint of the 1994 edition.

    Google Scholar 

  208. G. Iooss, Bifurcation of invariant tori in r3, Bifurcation and Nonlinear Eigenvalue Problems (C. Bardos, J. M. Lasry, and M. Schatzman, eds.), Lecture Notes in Mathematics, vol. 782, Springer Berlin Heidelberg, 1980, pp. 192–200.

    Google Scholar 

  209. M. C. Irwin, Smooth dynamical systems, Advanced Series in Nonlinear Dynamics, vol. 17, World Scientific Publishing Co., Inc., River Edge, NJ, 2001, Reprint of the 1980 original, With a foreword by R. S. MacKay.

    Google Scholar 

  210. T. H. Jäger, The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations, Mem. Amer. Math. Soc. 201 (2009), no. 945, vi+106.

    Google Scholar 

  211. T. H. Jäger, The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations, Mem. Amer. Math. Soc. 201 (2009), no. 945, vi+106.

    Google Scholar 

  212. T. H. Jäger, Strange non-chaotic attractors in quasiperiodically forced circle maps, Comm. Math. Phys. 289 (2009), no. 1, 253–289.

    Article  MathSciNet  MATH  Google Scholar 

  213. A. Jorba and R. de la Llave, Regularity properties of center manifolds and applications, manuscript.

    Google Scholar 

  214. À. Jorba, R. de la Llave, and M. Zou, Lindstedt series for lower-dimensional tori, Hamiltonian systems with three or more degrees of freedom (S’Agaró, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 533, Kluwer Acad. Publ., Dordrecht, 1999, pp. 151–167.

    Google Scholar 

  215. M. E. Johnson, M. S. Jolly, and I. G. Kevrekidis, Two-dimensional invariant manifolds and global bifurcations: some approximation and visualization studies, Numer. Algorithms 14 (1997), no. 1–3, 125–140, Dynamical numerical analysis (Atlanta, GA, 1995).

    Google Scholar 

  216. J. Jarník and J. Kurzweil, On invariant sets and invariant manifolds of differential systems, J. Differential Equations 6 (1969), 247–263.

    Article  MathSciNet  MATH  Google Scholar 

  217. A. Y. Jalnine, S. P. Kuznetsov, and A. H. Osbaldestin, Dynamics of small perturbations of orbits on a torus in a quasiperiodically forced 2D dissipative map, Regul. Chaotic Dyn. 11 (2006), no. 1, 19–30.

    Article  MathSciNet  MATH  Google Scholar 

  218. À. Jorba and J. Masdemont, Dynamics in the center manifold of the collinear points of the restricted three body problem, Phys. D 132 (1999), no. 1–2, 189–213.

    Article  MathSciNet  MATH  Google Scholar 

  219. 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 

  220. A. Y. Jalnine and A. H. Osbaldestin, Smooth and nonsmooth dependence of Lyapunov vectors upon the angle variable on a torus in the context of torus-doubling transitions in the quasiperiodically forced Hénon map, Phys. Rev. E (3) 71 (2005), no. 1, 016206, 14.

    Google Scholar 

  221. À. Jorba and E. Olmedo, On the computation of reducible invariant tori on a parallel computer, SIAM J. Appl. Dyn. Syst. 8 (2009), no. 4, 1382–1404.

    Article  MathSciNet  MATH  Google Scholar 

  222. À. Jorba, A methodology for the numerical computation of normal forms, centre manifolds and first integrals of Hamiltonian systems, Experiment. Math. 8 (1999), no. 2, 155–195.

    Article  MathSciNet  MATH  Google Scholar 

  223. A. Y. Jalnine and A. H. Osbaldestin, Numerical computation of the normal behaviour of invariant curves of n-dimensional maps, Nonlinearity 14 (2001), no. 5, 943–976.

    Article  MathSciNet  Google Scholar 

  224. R. A. Johnson and G. R. Sell, Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems, J. Differential Equations 41 (1981), no. 2, 262–288.

    Article  MathSciNet  MATH  Google Scholar 

  225. À. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differential Equations 98 (1992), no. 1, 111–124.

    Article  MathSciNet  MATH  Google Scholar 

  226. R. A. Johnson and G. R. Sell, On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal. 27 (1996), no. 6, 1704–1737.

    Article  MathSciNet  Google Scholar 

  227. À. Jorba and J. C. Tatjer, A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps, Discrete Contin. Dyn. Syst. Ser. B 10 (2008), no. 2–3, 537–567.

    MathSciNet  MATH  Google Scholar 

  228. À. Jorba, J. C. Tatjer, C. Núñez, and R. Obaya, Old and new results on strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 17 (2007), no. 11, 3895–3928.

    Article  MathSciNet  MATH  Google Scholar 

  229. I. Jungreis, A method for proving that monotone twist maps have no invariant circles, Ergodic Theory Dynam. Systems 11 (1991), no. 1, 79–84.

    Article  MathSciNet  MATH  Google Scholar 

  230. À. Jorba and J. Villanueva, On the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systems, Nonlinearity 10 (1997), no. 4, 783–822.

    Article  MathSciNet  MATH  Google Scholar 

  231. R. A. Johnson and G. R. Sell, On the persistence of lower-dimensional invariant tori under quasi-periodic perturbations, J. Nonlinear Sci. 7 (1997), no. 5, 427–473.

    Article  MathSciNet  Google Scholar 

  232. R. A. Johnson and G. R. Sell, Numerical computation of normal forms around some periodic orbits of the restricted three-body problem, Phys. D 114 (1998), no. 3–4, 197–229.

    MathSciNet  Google Scholar 

  233. À. Jorba and M. Zou, A software package for the numerical integration of ODEs by means of high-order Taylor methods, Experiment. Math. 14 (2005), no. 1, 99–117.

    Article  MathSciNet  MATH  Google Scholar 

  234. L. Kantorovitch and G. Akilov, Analyse fonctionnelle. Tome 2, “Mir”, Moscow, 1981, Équations fonctionnelles. [Functional equations], Translated from the second Russian edition by Djilali Embarek.

    Google Scholar 

  235. K. Kaneko, Fractalization of torus, Progr. Theoret. Phys. 71 (1984), no. 5, 1112–1115.

    Article  MathSciNet  MATH  Google Scholar 

  236. R. A. Johnson and G. R. Sell, Collapse of tori and genesis of chaos in dissipative systems, World Scientific Publishing Co., Singapore, 1986.

    Google Scholar 

  237. T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

    Google Scholar 

  238. 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 

  239. G. Kedem, Automatic differentiation of computer programs, ACM Trans. Math. Software 6 (1980), no. 2, 150–165.

    Article  MathSciNet  MATH  Google Scholar 

  240. A. Kelley, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations 3 (1967), 546–570.

    Article  MathSciNet  MATH  Google Scholar 

  241. G. Keller, A note on strange nonchaotic attractors, Fund. Math. 151 (1996), no. 2, 139–148.

    MathSciNet  MATH  Google Scholar 

  242. A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995, With a supplementary chapter by Katok and Leonardo Mendoza.

    Google Scholar 

  243. S.-Y. Kim and W. Lim, Universal mechanism for the intermittent route to strange nonchaotic attractors in quasiperiodically forced systems, Journal of Physics A: Mathematical and General 37 (2004), no. 25, 6477.

    Google Scholar 

  244. W. S. Koon, M. W. Lo, J. E. Marsden, and S. D. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos 10 (2000), no. 2, 427–469.

    Article  MathSciNet  MATH  Google Scholar 

  245. Yu. A. Kuznetsov and H. G. E. Meijer, Numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues, SIAM J. Sci. Comput. 26 (2005), no. 6, 1932–1954 (electronic).

    Google Scholar 

  246. T. Kaczynski, K. Mischaikow, and M. Mrozek, Computational homology, Applied Mathematical Sciences, vol. 157, Springer-Verlag, New York, 2004.

    MATH  Google Scholar 

  247. D. E. Knuth, The art of computer programming. Vol. 2: Seminumerical algorithms, third revised ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1997.

    Google Scholar 

  248. A. Karatsuba and Yu Ofman, Multiplication of many-digital numbers by automatic computers, Dokl. Akad. Nauk SSSR 145 (1962), 293–294, Translation in Physics-Doklady, 7 (1963), 595–596.

    Google Scholar 

  249. B. Krauskopf and H. M. Osinga, Globalizing two-dimensional unstable manifolds of maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 8 (1998), no. 3, 483–503.

    Article  MathSciNet  MATH  Google Scholar 

  250. H. Koch, A renormalization group fixed point associated with the breakup of golden invariant tori, Discrete Contin. Dyn. Syst. 11 (2004), no. 4, 881–909.

    Article  MathSciNet  MATH  Google Scholar 

  251. D. E. Knuth, Existence of critical invariant tori, Ergodic Theory Dynam. Systems 28 (2008), no. 6, 1879–1894.

    Article  MathSciNet  MATH  Google Scholar 

  252. B. Krauskopf, H. M. Osinga, E. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 (2005), no. 3, 763–791.

    Article  MathSciNet  MATH  Google Scholar 

  253. 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 

  254. A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR (N.S.) 98 (1954), 527–530, Translated in p. 51–56 of Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Como 1977 (eds. G. Casati and J. Ford) Lect. Notes Phys. 93, Springer, Berlin, 1979.

    Google Scholar 

  255. J. A. Ketoja and I. I. Satija, Self-Similarity and Localization, Phys. Rev. Lett. 75 (1995), no. 14, 2762–2765.

    Article  Google Scholar 

  256. J. A. Ketoja and I. I. Satija, Harper equation, the dissipative standard map and strange nonchaotic attractors: relationship between an eigenvalue problem and iterated maps, Phys. D 109 (1997), no. 1–2, 70–80, Physics and dynamics between chaos, order, and noise (Berlin, 1996).

    Google Scholar 

  257. H. Koch, A. Schenkel, and P. Wittwer, Computer-assisted proofs in analysis and programming in logic: a case study, SIAM Rev. 38 (1996), no. 4, 565–604.

    Article  MathSciNet  MATH  Google Scholar 

  258. S. B. Kuksin, Perturbation of conditionally periodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 1, 41–63, 240, Translated in Math. USSR-Izv., 32(1): 39–62, 1989.

    Google Scholar 

  259. S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, vol. 19, Oxford University Press, 2000.

    Google Scholar 

  260. W. T. Kyner, Invariant manifolds, Rend. Circ. Mat. Palermo (2) 10 (1961), 98–110.

    Google Scholar 

  261. O. E. Lanford, III, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 427–434.

    Google Scholar 

  262. O. E. Lanford, Computer-assisted proofs in analysis, Phys. A 124 (1984), no. 1–3, 465–470, Mathematical physics, VII (Boulder, Colo., 1983).

    Google Scholar 

  263. O. E. Lanford, A shorter proof of the existence of the Feigenbaum fixed point, Comm. Math. Phys. 96 (1984), no. 4, 521–538.

    Article  MathSciNet  MATH  Google Scholar 

  264. J. Laskar, Manipulation des séries, 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), 89–108.

    Google Scholar 

  265. O. E. Lanford, Frequency map analysis and quasiperiodic decompositions, Hamiltonian systems and Fourier analysis, Adv. Astron. Astrophys., Camb. Sci. Publ., Cambridge, 2005, pp. 99–133.

    Google Scholar 

  266. V. F. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions, Springer-Verlag, Berlin, 1993.

    Book  MATH  Google Scholar 

  267. U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems, Cel. Mech. 78 (2000), no. 1, 47–74.

    Article  MathSciNet  MATH  Google Scholar 

  268. V. F. Lazutkin, Construction of Kolmogorov’s normal form for a planetary system, Regul. Chaotic Dyn. 10 (2005), no. 2, 153–171.

    Article  MathSciNet  Google Scholar 

  269. 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 

  270. J.-P. Lessard, J. D. Mireles James, and C. Reinhardt, Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields, J. Dynam. Differential Equations 26 (2014), no. 2, 267–313.

    Article  MathSciNet  MATH  Google Scholar 

  271. M. W. Lo, The Interplanetary Superhighway and the origins program, 2002 IEEE Aerospace Conference Proceedings, Vols 1–7, 2002, pp. 3543–3562.

    Google Scholar 

  272. U. Locatelli, Three-body planetary problem: study of KAM stability for the secular part of the Hamiltonian, Planetary and Space Science 46 (1998), no. 11, 1453–1464.

    Article  Google Scholar 

  273. E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences 20 (1963), no. 2, 130–141.

    Article  Google Scholar 

  274. Yu. D. Latushkin and A. M. Stëpin, Weighted shift operators, the spectral theory of linear extensions and a multiplicative ergodic theorem, Mat. Sb. 181 (1990), no. 6, 723–742.

    Google Scholar 

  275. A. Luque and J. Villanueva, Computation of derivatives of the rotation number for parametric families of circle diffeomorphisms, Phys. D 237 (2008), no. 20, 2599–2615.

    Article  MathSciNet  MATH  Google Scholar 

  276. Yu. D. Latushkin and A. M. Stëpin, Numerical computation of rotation numbers for quasi-periodic planar curves, Phys. D 238 (2009), no. 20, 2025–2044.

    Article  MathSciNet  Google Scholar 

  277. Yu. D. Latushkin and A. M. Stëpin, A KAM theorem without action-angle variables for elliptic lower dimensional tori, Nonlinearity 24 (2011), no. 4, 1033–1080.

    Article  MathSciNet  Google Scholar 

  278. Yu. D. Latushkin and A. M. Stëpin, Quasi-periodic frequency analysis using averaging-extrapolation methods, SIAM J. Appl. Dyn. Syst. 13 (2014), no. 1, 1–46.

    Article  MathSciNet  Google Scholar 

  279. A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control 55 (1992), no. 3, 521–790, Translated by A. T. Fuller from Édouard Davaux’s French translation (1907) of the 1892 Russian original, With an editorial (historical introduction) by Fuller, a biography of Lyapunov by V. I. Smirnov, and the bibliography of Lyapunov’s works collected by J. F. Barrett, Lyapunov centenary issue.

    Google Scholar 

  280. R. S. MacKay, Renormalisation in area-preserving maps, Advanced Series in Nonlinear Dynamics, vol. 6, World Scientific Publishing Co. Inc., River Edge, NJ, 1993.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  283. J. N. Mather, Non-existence of invariant circles, Ergodic Theory Dyn. Syst. 4 (1984), 301–309.

    Article  MathSciNet  MATH  Google Scholar 

  284. K. Makino and M. Berz, Taylor models and other validated functional inclusion methods, Int. J. Pure Appl. Math. 4 (2003), no. 4, 379–456.

    Google Scholar 

  285. 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 

  286. 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 

  287. V. K. Melnikov, On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function, Soviet Math. Dokl. 6 (1965), no. 6, 1592–1596.

    Google Scholar 

  288. V. K. Melnikov, A family of conditionally periodic solutions of a Hamiltonian systems, Soviet Math. Dokl. 9 (1968), 882–886.

    Google Scholar 

  289. J. Milnor, On the concept of attractor, Commun. Math. Phys. 99 (1985), 177–195.

    Article  MathSciNet  MATH  Google Scholar 

  290. J. D. Mireles James, Quadratic volume-preserving maps: (un)stable manifolds, hyperbolic dynamics, and vortex-bubble bifurcations, J. Nonlinear Sci. 23 (2013), no. 4, 585–615.

    Article  MathSciNet  MATH  Google Scholar 

  291. J. D. Mireles James, Computer assisted error bounds for linear approximation of (un)stable manifolds and rigorous validation of higher dimensional transverse connecting orbits, Commun. Nonlinear Sci. Numer. Simul. 22 (2015), no. 1–3, 1102–1133.

    Article  MathSciNet  MATH  Google Scholar 

  292. J. D. Mireles James, Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds, Indag. Math. (N.S.) 26 (2015), no. 1, 225–265.

    Google Scholar 

  293. 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), no. 3, 919–953.

    Article  MathSciNet  MATH  Google Scholar 

  294. J. D. Mireles James and K. Mischaikow, Rigorous a posteriori computation of (un)stable manifolds and connecting orbits for analytic maps, SIAM J. Appl. Dyn. Syst. 12 (2013), no. 2, 957–1006.

    Article  MathSciNet  MATH  Google Scholar 

  295. J. D. Mireles James and, Computational proofs in dynamics, Encyclopedia of Applied Computational Mathematics (E. Björn, ed.), Springer-Verlag, Berlin, 2015, p. 1676.

    Google Scholar 

  296. 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 

  297. 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 

  298. R. E. Moore, Interval analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1966.

    MATH  Google Scholar 

  299. R. E. Moore, Methods and applications of interval analysis, SIAM Studies in Applied Mathematics, vol. 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1979.

    Google Scholar 

  300. 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 

  301. 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 

  302. J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 (1962), 1–20.

    MathSciNet  MATH  Google Scholar 

  303. R. E. Moore, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294.

    Article  MathSciNet  Google Scholar 

  304. R. E. Moore, On the theory of quasiperiodic motions, SIAM Rev. 8 (1966), no. 2, 145–172.

    Article  MathSciNet  Google Scholar 

  305. R. E. Moore, A rapidly convergent iteration method and non-linear differential equations. II, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 499–535.

    Google Scholar 

  306. R. E. Moore, A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 265–315.

    Google Scholar 

  307. R. E. Moore, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136–176.

    Article  MathSciNet  MATH  Google Scholar 

  308. R. E. Moore, On a theorem of Anosov, J. Differential Equations 5 (1969), 411–440.

    Article  MathSciNet  Google Scholar 

  309. B. D. Mestel, A. H. Osbaldestin, and B. Winn, Golden mean renormalization for the Harper equation: the strong coupling fixed point, J. Math. Phys. 41 (2000), no. 12, 8304–8330.

    Article  MathSciNet  MATH  Google Scholar 

  310. R. S. MacKay and I. C. Percival, Converse KAM: theory and practice, Comm. Math. Phys. 98 (1985), no. 4, 469–512.

    Article  MathSciNet  MATH  Google Scholar 

  311. K. R. Meyer and D. S. Schmidt (eds.), Computer aided proofs in analysis, The IMA Volumes in Mathematics and its Applications, vol. 28, New York, Springer-Verlag, 1991.

    Google Scholar 

  312. D. McDuff and D. Salamon, Introduction to symplectic topology, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998.

    Google Scholar 

  313. R. Martínez and C. Simó, Invariant manifolds at infinity of the RTBP and the boundaries of bounded motion, Regul. Chaotic Dyn. 19 (2014), no. 6, 745–765.

    Article  MathSciNet  MATH  Google Scholar 

  314. J. Murdock, Normal forms and unfoldings for local dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.

    Book  MATH  Google Scholar 

  315. S. Newhouse, M. Berz, J. Grote, and K. Makino, On the estimation of topological entropy on surfaces, Geometric and probabilistic structures in dynamics, Contemp. Math., vol. 469, Amer. Math. Soc., Providence, RI, 2008, pp. 243–270.

    Google Scholar 

  316. R. D. Neidinger, Computing multivariable Taylor series to arbitrary order, SIGAPL APL Quote Quad 25 (1995), no. 4, 134–144.

    Article  Google Scholar 

  317. R. D. Neidinger, Directions for computing truncated multivariate Taylor series, Math. Comp. 74 (2005), no. 249, 321–340 (electronic).

    Google Scholar 

  318. R. D. Neidinger, Introduction to automatic differentiation and matlab object-oriented programming, SIAM Review 52 (2010), no. 3, 545–563.

    Article  MathSciNet  MATH  Google Scholar 

  319. T. Nishikawa and K. Kaneko, Fractalization of a torus as a strange nonchaotic attractor, Phys. Rev. E 54 (1996), no. 6, 6114–6124.

    Article  Google Scholar 

  320. A. Olvera and N. P. Petrov, Regularity properties of critical invariant circles of twist maps, and their universality, SIAM J. Appl. Dyn. Syst. 7 (2008), no. 3, 962–987.

    Article  MathSciNet  MATH  Google Scholar 

  321. M. Ollé, J. R. Pacha, and J. Villanueva, Kolmogorov-Arnold-Moser aspects of the periodic Hamiltonian Hopf bifurcation, Nonlinearity 21 (2008), no. 8, 1759–1811.

    Article  MathSciNet  MATH  Google Scholar 

  322. H. M. Osinga, J. Wiersig, P. Glendinning, and U. Feudel, Multistability and nonsmooth bifurcations in the quasiperiodically forced circle map, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 11 (2001), no. 12, 3085–3105.

    Article  MathSciNet  MATH  Google Scholar 

  323. J. Palis and W. de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982, An introduction, Translated from the Portuguese by A. K. Manning.

    Google Scholar 

  324. O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z. 29 (1929), no. 1, 129–160.

    Article  MathSciNet  MATH  Google Scholar 

  325. J. S. Parker and M. W. Lo, Shoot the moon 3D, Advances in the Astronautical Sciences 123 (2006), 2067–2086.

    Google Scholar 

  326. V. A. Pliss, A reduction principle in the theory of stability of motion, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1297–1324.

    MathSciNet  MATH  Google Scholar 

  327. H. Poincaré, Sur une classe nouvelle de transcendentes uniformes, Jour. de Math. 6 (1890), 313–365.

    Google Scholar 

  328. V. A. Pliss, Les méthodes nouvelles de la mécanique céleste. Tome II, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Librairie Scientifique et Technique Albert Blanchard, Paris, 1987.

    Google Scholar 

  329. J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 35 (1982), no. 5, 653–696.

    Article  MathSciNet  MATH  Google Scholar 

  330. V. A. Pliss, On invariant manifolds of complex analytic mappings near fixed points, Exposition. Math. 4 (1986), no. 2, 97–109.

    MathSciNet  Google Scholar 

  331. V. A. Pliss, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z. 202 (1989), no. 4, 559–608.

    Article  MathSciNet  Google Scholar 

  332. V. A. Pliss, A lecture on the classical KAM theorem, Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 707–732.

    Google Scholar 

  333. 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 

  334. L. B. Rall, Differentiation and generation of Taylor coefficients in pascal-sc, A new approach to scientific computation. Proceedings of the symposium held at the IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y., August 3, 1982 (New York) (U.W. Kulisch and W.L. Miranker, eds.), Notes and Reports in Computer Science and Applied Mathematics, vol. 7, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1983, pp. xv+384.

    Google Scholar 

  335. 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 

  336. 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 

  337. N. Revol, K. Makino, and M. Berz, Taylor models and floating-point arithmetic: proof that arithmetic operations are validated in COSY, J. Log. Algebr. Program. 64 (2005), no. 1, 135–154.

    Article  MathSciNet  MATH  Google Scholar 

  338. C. Robinson, Dynamical systems, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995, Stability, symbolic dynamics, and chaos.

    Google Scholar 

  339. L. B. Rall, An introduction to dynamical systems: continuous and discrete, Pearson Prentice Hall, Upper Saddle River, NJ, 2004.

    Google Scholar 

  340. A. Rom, Mechanized algebraic operations (mao), Celestial mechanics 1 (1970), no. 3–4, 301–319 (English).

    Article  MathSciNet  MATH  Google Scholar 

  341. N. Revol and F. Rouillier, Motivations for an arbitrary precision interval arithmetic and the mpfi library, Reliable computing 11 (2005), no. 5, 275–290.

    Article  MathSciNet  MATH  Google Scholar 

  342. H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Springer, Berlin, 1975, pp. 598–624. Lecture Notes in Phys., Vol. 38.

    Google Scholar 

  343. L. B. Rall, On a new proof of Moser’s twist mapping theorem, Proceedings of the Fifth Conference on Mathematical Methods in Celestial Mechanics (Oberwolfach, 1975), Part I. Celestial Mech., 14(1):19–31, 1976.

    Article  Google Scholar 

  344. L. B. Rall, On optimal estimates for the solutions of linear difference equations on the circle, Proceedings of the Fifth Conference on Mathematical Methods in Celestial Mechanics (Oberwolfach, 1975), Part I. Celestial Mech., vol. 14, 1976.

    Google Scholar 

  345. L. B. Rall, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn. 6 (2001), no. 2, 119–204.

    Article  MathSciNet  Google Scholar 

  346. R. J. Sacker, A new approach to the perturbation theory of invariant surfaces, Comm. Pure Appl. Math. 18 (1965), 717–732.

    Article  MathSciNet  MATH  Google Scholar 

  347. R. J. Sacker, A perturbation theorem for invariant manifolds and Hölder continuity, J. Math. Mech. 18 (1969), 705–762.

    MathSciNet  MATH  Google Scholar 

  348. É. Schost, Multivariate power series multiplication, ISSAC’05, ACM, New York, 2005, pp. 293–300 (electronic).

    Google Scholar 

  349. M. B. Sevryuk, Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman’s method, Discrete Contin. Dyn. Syst. 18 (2007), no. 2–3, 569–595.

    Article  MathSciNet  MATH  Google Scholar 

  350. O. Sosnovtseva, U. Feudel, J. Kurths, and A. Pikovsky, Multiband strange nonchaotic attractors in quasiperiodically forced systems, Physics Letters A 218 (1996), no. 3–6, 255–267.

    Article  Google Scholar 

  351. J. Sijbrand, Properties of center manifolds, Trans. Amer. Math. Soc. 289 (1985), no. 2, 431–469.

    Article  MathSciNet  MATH  Google Scholar 

  352. 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 

  353. M. B. Sevryuk, Effective computations in Hamiltonian dynamics, Mécanique céleste, SMF Journ. Annu., vol. 1996, Soc. Math. France, Paris, 1996, p. 23.

    Google Scholar 

  354. M. B. Sevryuk, Effective computations in celestial mechanics and astrodynamics, Modern methods of analytical mechanics and their applications (Udine, 1997), CISM Courses and Lectures, vol. 387, Springer, Vienna, 1998, pp. 55–102.

    Google Scholar 

  355. M. B. Sevryuk, Global dynamics and fast indicators, Global analysis of dynamical systems, Inst. Phys., Bristol, 2001, pp. 373–389.

    Google Scholar 

  356. M. B. Sevryuk, On the role of dynamical systems in celestial mechanics. Accessed March 14, 2016, http://www.maia.ub.es/dsg/2010/1004simo.ps.gz.

    Google Scholar 

  357. C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer-Verlag, New York, 1971, Translation by C. I. Kalme, Die Grundlehren der mathematischen Wissenschaften, Band 187.

    Google Scholar 

  358. 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 

  359. 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 

  360. C. Sparrow, The Lorenz equations: bifurcations, chaos, and strange attractors, Applied Mathematical Sciences, vol. 41, Springer-Verlag, New York-Berlin, 1982.

    Book  MATH  Google Scholar 

  361. A. Schönhage and V. Strassen, Schnelle Multiplikation grosser Zahlen, Computing (Arch. Elektron. Rechnen) 7 (1971), 281–292.

    Google Scholar 

  362. R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations 15 (1974), 429–458.

    Article  MathSciNet  MATH  Google Scholar 

  363. B. Sandstede and T. Theerakarn, Regularity of center manifolds via the graph transform, Journal of Dynamics and Differential Equations (2015), 1–18 (English).

    Google Scholar 

  364. J. Stark, Invariant graphs for forced systems, Phys. D 109 (1997), no. 1–2, 163–179, Physics and dynamics between chaos, order, and noise (Berlin, 1996).

    Google Scholar 

  365. E. M. Standish, JPL planetary and lunar ephemerides, DE405/LE405, Tech. Report IOM 312.F.98–048, Jet Propultion Laboratory, 1998.

    Google Scholar 

  366. J. Stark, Regularity of invariant graphs for forced systems, Ergodic Theory Dynam. Systems 19 (1999), no. 1, 155–199.

    Article  MathSciNet  MATH  Google Scholar 

  367. S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809–824.

    Article  MathSciNet  MATH  Google Scholar 

  368. E. M. Standish, On the structure of local homeomorphisms of euclidean n-space. II., Amer. J. Math. 80 (1958), 623–631.

    Article  MathSciNet  Google Scholar 

  369. T. M. Seara and J. Villanueva, On the numerical computation of Diophantine rotation numbers of analytic circle maps, Phys. D 217 (2006), no. 2, 107–120.

    Article  MathSciNet  MATH  Google Scholar 

  370. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  372. D. Salamon and E. Zehnder, KAM theory in configuration space, Comment. Math. Helv. 64 (1989), no. 1, 84–132.

    Article  MathSciNet  MATH  Google Scholar 

  373. V. Szebehely, Theory of orbits. the Restricted Problem of Three Bodies, Academic Press, 1967.

    Google Scholar 

  374. F. Takens, Normal forms for certain singularities of vectorfields, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 2, 163–195, Colloque International sur l’Analyse et la Topologie Différentielle (Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1972).

    Google Scholar 

  375. S. Tompaidis, Approximation of invariant surfaces by periodic orbits in high-dimensional maps: some rigorous results, Experiment. Math. 5 (1996), no. 3, 197–209.

    Article  MathSciNet  MATH  Google Scholar 

  376. A. L. Toom, The complexity of a scheme of functional elements simulating the multiplication of integers, Dokl. Akad. Nauk SSSR 150 (1963), 496–498.

    MathSciNet  MATH  Google Scholar 

  377. 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 

  378. W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 12, 1197–1202.

    Article  MathSciNet  MATH  Google Scholar 

  379. M. R. Trummer, A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math. 2 (2002), no. 1, 53–117.

    Article  MathSciNet  MATH  Google Scholar 

  380. M. R. Trummer, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011.

    Google Scholar 

  381. T. Uzer, C. Jaffé, J. Palacián, P. Yanguas, and S. Wiggins, The geometry of reaction dynamics, Nonlinearity 15 (2002), no. 4, 957–992.

    Article  MathSciNet  MATH  Google Scholar 

  382. E. Valdinoci, Families of whiskered tori for a-priori stable/unstable Hamiltonian systems and construction of unstable orbits, Math. Phys. Electron. J. 6 (2000), Paper 2, 31 pp. (electronic).

    Google Scholar 

  383. R. Vitolo, H. Broer, and C. Simó, Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems, Regul. Chaotic Dyn. 16 (2011), no. 1–2, 154–184.

    Article  MathSciNet  MATH  Google Scholar 

  384. J. B. van den Berg, J. D. Mireles James, J.-P. Lessard, and K. Mischaikow, Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray-Scott equation, SIAM J. Math. Anal. 43 (2011), no. 4, 1557–1594.

    Article  MathSciNet  MATH  Google Scholar 

  385. J. van der Hoeven, Relax, but don’t be too lazy, J. Symbolic Comput. 34 (2002), no. 6, 479–542.

    Article  MathSciNet  MATH  Google Scholar 

  386. J. B. van den Berg, J. D. Mireles James, J.-P. Lessard, and K., The truncated Fourier transform and applications, ISSAC 2004, ACM, New York, 2004, pp. 290–296.

    Google Scholar 

  387. J. Villanueva, Kolmogorov theorem revisited, J. Differential Equations 244 (2008), no. 9, 2251–2276.

    Article  MathSciNet  MATH  Google Scholar 

  388. 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 

  389. A. Weinstein, Lectures on symplectic manifolds, CBMS Regional Conf. Ser. in Math., vol. 29, Amer. Math. Soc., Providence, 1977.

    Google Scholar 

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

    Book  MATH  Google Scholar 

  391. A. Wittig, Rigorous high-precision enclosures of fixed points and their invariant manifolds, Ph.D. thesis, Department of Physics and Astronomy, Michigan State University, 2011.

    Google Scholar 

  392. J. Xu and J. You, Persistence of lower-dimensional tori under the first Melnikov’s non-resonance condition, J. Math. Pures Appl. (9) 80 (2001), no. 10, 1045–1067.

    Google Scholar 

  393. J. Xu, J. You, and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z. 226 (1997), no. 3, 375–387.

    Article  MathSciNet  MATH  Google Scholar 

  394. E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math. 28 (1975), 91–140.

    Article  MathSciNet  MATH  Google Scholar 

  395. J. B. van den Berg, J. D. Mireles James, J.-P. Lessard, and K., Generalized implicit function theorems with applications to some small divisor problems. II, Comm. Pure Appl. Math. 29 (1976), no. 1, 49–111.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Jordi-Lluís Figueras

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

    Àlex Haro

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Figueras, JL., Haro, À. (2016). The Parameterization Method for Quasi-Periodic Systems: From Rigorous Results to Validated Numerics. In: The Parameterization Method for Invariant Manifolds. Applied Mathematical Sciences, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-29662-3_3

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