# American Institute of Mathematical Sciences

April  2012, 32(4): 1309-1353. doi: 10.3934/dcds.2012.32.1309

## Computation of whiskered invariant tori and their associated manifolds: New fast algorithms

 1 Center for Neural Science, New York University, New York, NY 10003, United States 2 Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712-1082, United States 3 Université Paul Cézanne, Laboratoire LATP UMR 6632, Marseille, France

Received  September 2010 Revised  August 2011 Published  October 2011

We present efficient (low storage requirement and low operation count) algorithms for the computation of several invariant objects for Hamiltonian dynamics, namely KAM tori (i.e diffeomorphic copies of tori such that the motion on them is conjugated to a rigid rotation) both Lagrangian tori(of maximal dimension) and whiskered tori (i.e. tori with hyperbolic directions which, together with the tangents to the torus and the symplectic conjugates span the whole tangent space). We also present algorithms to compute the invariant splitting and the invariant manifolds of whiskered tori. We present the algorithms for both discrete-time dynamical systems and differential equations.
The algorithms do not require that the system is presented in action-angle variables nor that it is close to integrable and are backed up by rigorous a-posteriori bounds. We will report on the implementation results elsewhere.
Citation: Gemma Huguet, Rafael de la Llave, Yannick Sire. Computation of whiskered invariant tori and their associated manifolds: New fast algorithms. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1309-1353. doi: 10.3934/dcds.2012.32.1309
##### References:
 [1] V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady, 5 (1964), 581-585. Google Scholar [2] S. Aubry, The twist map, the extended Frenkel-Kontorova model and the devil's staircase. Order in chaos, (Los Alamos, N.M., 1982), Phys. D, 7 (1983), 240-258. doi: 10.1016/0167-2789(83)90129-X.  Google Scholar [3] S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Phys. D, 8 (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6.  Google Scholar [4] A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Mem. Amer. Math. Soc., 187 (2007), viii+134 pp.  Google Scholar [5] A. Celletti, C. Falcolini and U. Locatelli, On the break-down threshold of invariant tori in four dimensional maps, Regul. Chaotic Dyn., 9 (2004), 227-253. doi: 10.1070/RD2004v009n03ABEH000278.  Google Scholar [6] 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), 1311-1336. doi: 10.1088/0951-7715/22/6/004.  Google Scholar [7] R. Calleja and R. de la Llave, A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification, Nonlinearity, 23 (2010), 2029-2058. doi: 10.1088/0951-7715/23/9/001.  Google Scholar [8] R. Calleja and R. de la Llave, Computation of the breakdown of analyticity in statistical mechanics models: Numerical results and a renormalization group explanation, J. Stat. Phys, 141 (2010), 940-951. doi: 10.1007/s10955-010-0085-7.  Google Scholar [9] 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), 283-328. doi: 10.1512/iumj.2003.52.2245.  Google Scholar [10] X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), 329-360. doi: 10.1512/iumj.2003.52.2407.  Google Scholar [11] X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. III. Overview and applications, J. Differential Equations, 218 (2005), 444-515. doi: 10.1016/j.jde.2004.12.003.  Google Scholar [12] A. Delshams and G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity, 22 (2009), 1997-2077. doi: 10.1088/0951-7715/22/8/013.  Google Scholar [13] 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), viii+141 pp.  Google Scholar [14] P Duarte, Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 359-409.  Google Scholar [15] 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), 3136-3213. doi: 10.1016/j.jde.2009.01.037.  Google Scholar [16] E. Fontich, R. de la Llave and Y. Sire, A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems, Electron. Res. Announc. Math. Sci., 16 (2009), 9-22.  Google Scholar [17] F. Fassò, M. Guzzo and G. Benettin, Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, Comm. Math. Phys., 197 (1998), 347-360. doi: 10.1007/s002200050454.  Google Scholar [18] M. Frigo and S. G. Johnson, The design and implementation of FFTW3, Proceedings of the IEEE, 93 (2005), 216-231. doi: 10.1109/JPROC.2004.840301.  Google Scholar [19] B. Fayad, A. Katok and A. Windsor, Mixed spectrum reparameterizations of linear flows on $\mathbbT^ 2$, Mosc. Math. J., 1 (2001), 521-537, 644.  Google Scholar [20] M. Guzzo, F. Fassò and G. Benettin, On the stability of elliptic equilibria, Math. Phys. Electron. J., 4 (1998), Paper 1, 16 pp. (electronic).  Google Scholar [21] Samuel M. Graff, On the conservation of hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations, 15 (1974), 1-69. doi: 10.1016/0022-0396(74)90086-2.  Google Scholar [22] J. M. Greene, A method for determining a stochastic transition, Jour. Math. Phys., 20 (1979), 1183-1201. doi: 10.1063/1.524170.  Google Scholar [23] G. H. Golub and C. F. Van Loan, "Matrix Computations,'' Third edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996.  Google Scholar [24] Alex Haro, Automatic differentiation tools in computational dynamical systems, Manuscript, 2008. Google Scholar [25] G. Huguet, R. de la Llave and Y. Sire, Fast iteration of cocyles over rotations and Computation of hyperbolic bundles,, preprint, ().   Google Scholar [26] Michael-R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau. Vol. 1, With an appendix by Albert Fathi, With an English summary, in "Astérisque," 103-104, Société Mathématique de France, Paris, 1983.  Google Scholar [27] M.-R. Herman, On the dynamics of Lagrangian tori invariant by symplectic diffeomorphisms, in "Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations" (L'Aquila, 1990), Pitman Res. Notes Math. Ser., 243, Longman Sci. Tech., Harlow, (1992), 92-112.  Google Scholar [28] A. Haro and R. de la Llave, New mechanisms for lack of equipartion of energy, Phys. Rev. Lett., 89 (2000), 1859-1862. doi: 10.1103/PhysRevLett.85.1859.  Google Scholar [29] À. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8 pp.  Google Scholar [30] À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300. doi: 10.3934/dcdsb.2006.6.1261.  Google Scholar [31] À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results, J. Differential Equations, 228 (2006), 530-579. doi: 10.1016/j.jde.2005.10.005.  Google Scholar [32] À. Haro and R. de la Llave, 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 Jour. Appl. Dyn. Syst., 6 (2007), 142-207. doi: 10.1137/050637327.  Google Scholar [33] À. Jorba and E. Olmedo, A parallel method to compute quasi-periodic solutions, in "EQUADIFF 2003," 181-183, World Sci. Publ., Hackensack, NJ, 2005.  Google Scholar [34] À. Jorba and E. Olmedo, On the computation of reducible invariant tori on a parallel computer, SIAM J. Appl. Dyn. Syst., 8 (2009), 1382-1404. doi: 10.1137/080724563.  Google Scholar [35] D. E. Knuth, "The Art of Computer Programming. Vol. 2: Seminumerical Algorithms,'' Third revised edition, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1997. Google Scholar [36] R. de la Llave, A. Gonzàlez, À. Jorba and J. Villanueva, KAM theory without action-angle variables, Nonlinearity, 18 (2005), 855-895. doi: 10.1088/0951-7715/18/2/020.  Google Scholar [37] R. de la Llave, A tutorial on KAM theory, in "Smooth Ergodic Theory and its Applications" (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, (2001), 175-292.  Google Scholar [38] 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 [39] R. S. McKay, "Renormalisation in Area Preserving Maps,'' Ph.D thesis, Princeton University, 1982. Google Scholar [40] 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), 962-987. doi: 10.1137/070687967.  Google Scholar [41] M. J. Raković and Shih-I Chu, New integrable systems: Hydrogen atom in external fields, Phys. D, 81 (1995), 271-279. doi: 10.1016/0167-2789(94)00220-K.  Google Scholar [42] M. J. Raković and Shih-I Chu, Phase-space structure of a new integrable system related to hydrogen atoms in external fields, J. Phys. A, 30 (1997), 733-753. doi: 10.1088/0305-4470/30/2/033.  Google Scholar [43] H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, in "Dynamical Systems, Theory and Applications" (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Phys., Vol. 38, Berlin, Springer, (1975), 598-624.  Google Scholar [44] C. Simó, private communication private communication, 2000. Google Scholar [45] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. II, Comm. Pure Appl. Math., 29 (1976), 49-111. doi: 10.1002/cpa.3160290104.  Google Scholar

show all references

##### References:
 [1] V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady, 5 (1964), 581-585. Google Scholar [2] S. Aubry, The twist map, the extended Frenkel-Kontorova model and the devil's staircase. Order in chaos, (Los Alamos, N.M., 1982), Phys. D, 7 (1983), 240-258. doi: 10.1016/0167-2789(83)90129-X.  Google Scholar [3] S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Phys. D, 8 (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6.  Google Scholar [4] A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Mem. Amer. Math. Soc., 187 (2007), viii+134 pp.  Google Scholar [5] A. Celletti, C. Falcolini and U. Locatelli, On the break-down threshold of invariant tori in four dimensional maps, Regul. Chaotic Dyn., 9 (2004), 227-253. doi: 10.1070/RD2004v009n03ABEH000278.  Google Scholar [6] 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), 1311-1336. doi: 10.1088/0951-7715/22/6/004.  Google Scholar [7] R. Calleja and R. de la Llave, A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification, Nonlinearity, 23 (2010), 2029-2058. doi: 10.1088/0951-7715/23/9/001.  Google Scholar [8] R. Calleja and R. de la Llave, Computation of the breakdown of analyticity in statistical mechanics models: Numerical results and a renormalization group explanation, J. Stat. Phys, 141 (2010), 940-951. doi: 10.1007/s10955-010-0085-7.  Google Scholar [9] 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), 283-328. doi: 10.1512/iumj.2003.52.2245.  Google Scholar [10] X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), 329-360. doi: 10.1512/iumj.2003.52.2407.  Google Scholar [11] X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. III. Overview and applications, J. Differential Equations, 218 (2005), 444-515. doi: 10.1016/j.jde.2004.12.003.  Google Scholar [12] A. Delshams and G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity, 22 (2009), 1997-2077. doi: 10.1088/0951-7715/22/8/013.  Google Scholar [13] 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), viii+141 pp.  Google Scholar [14] P Duarte, Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 359-409.  Google Scholar [15] 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), 3136-3213. doi: 10.1016/j.jde.2009.01.037.  Google Scholar [16] E. Fontich, R. de la Llave and Y. Sire, A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems, Electron. Res. Announc. Math. Sci., 16 (2009), 9-22.  Google Scholar [17] F. Fassò, M. Guzzo and G. Benettin, Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, Comm. Math. Phys., 197 (1998), 347-360. doi: 10.1007/s002200050454.  Google Scholar [18] M. Frigo and S. G. Johnson, The design and implementation of FFTW3, Proceedings of the IEEE, 93 (2005), 216-231. doi: 10.1109/JPROC.2004.840301.  Google Scholar [19] B. Fayad, A. Katok and A. Windsor, Mixed spectrum reparameterizations of linear flows on $\mathbbT^ 2$, Mosc. Math. J., 1 (2001), 521-537, 644.  Google Scholar [20] M. Guzzo, F. Fassò and G. Benettin, On the stability of elliptic equilibria, Math. Phys. Electron. J., 4 (1998), Paper 1, 16 pp. (electronic).  Google Scholar [21] Samuel M. Graff, On the conservation of hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations, 15 (1974), 1-69. doi: 10.1016/0022-0396(74)90086-2.  Google Scholar [22] J. M. Greene, A method for determining a stochastic transition, Jour. Math. Phys., 20 (1979), 1183-1201. doi: 10.1063/1.524170.  Google Scholar [23] G. H. Golub and C. F. Van Loan, "Matrix Computations,'' Third edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996.  Google Scholar [24] Alex Haro, Automatic differentiation tools in computational dynamical systems, Manuscript, 2008. Google Scholar [25] G. Huguet, R. de la Llave and Y. Sire, Fast iteration of cocyles over rotations and Computation of hyperbolic bundles,, preprint, ().   Google Scholar [26] Michael-R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau. Vol. 1, With an appendix by Albert Fathi, With an English summary, in "Astérisque," 103-104, Société Mathématique de France, Paris, 1983.  Google Scholar [27] M.-R. Herman, On the dynamics of Lagrangian tori invariant by symplectic diffeomorphisms, in "Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations" (L'Aquila, 1990), Pitman Res. Notes Math. Ser., 243, Longman Sci. Tech., Harlow, (1992), 92-112.  Google Scholar [28] A. Haro and R. de la Llave, New mechanisms for lack of equipartion of energy, Phys. Rev. Lett., 89 (2000), 1859-1862. doi: 10.1103/PhysRevLett.85.1859.  Google Scholar [29] À. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8 pp.  Google Scholar [30] À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300. doi: 10.3934/dcdsb.2006.6.1261.  Google Scholar [31] À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results, J. Differential Equations, 228 (2006), 530-579. doi: 10.1016/j.jde.2005.10.005.  Google Scholar [32] À. Haro and R. de la Llave, 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 Jour. Appl. Dyn. Syst., 6 (2007), 142-207. doi: 10.1137/050637327.  Google Scholar [33] À. Jorba and E. Olmedo, A parallel method to compute quasi-periodic solutions, in "EQUADIFF 2003," 181-183, World Sci. Publ., Hackensack, NJ, 2005.  Google Scholar [34] À. Jorba and E. Olmedo, On the computation of reducible invariant tori on a parallel computer, SIAM J. Appl. Dyn. Syst., 8 (2009), 1382-1404. doi: 10.1137/080724563.  Google Scholar [35] D. E. Knuth, "The Art of Computer Programming. Vol. 2: Seminumerical Algorithms,'' Third revised edition, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1997. Google Scholar [36] R. de la Llave, A. Gonzàlez, À. Jorba and J. Villanueva, KAM theory without action-angle variables, Nonlinearity, 18 (2005), 855-895. doi: 10.1088/0951-7715/18/2/020.  Google Scholar [37] R. de la Llave, A tutorial on KAM theory, in "Smooth Ergodic Theory and its Applications" (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, (2001), 175-292.  Google Scholar [38] 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 [39] R. S. McKay, "Renormalisation in Area Preserving Maps,'' Ph.D thesis, Princeton University, 1982. Google Scholar [40] 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), 962-987. doi: 10.1137/070687967.  Google Scholar [41] M. J. Raković and Shih-I Chu, New integrable systems: Hydrogen atom in external fields, Phys. D, 81 (1995), 271-279. doi: 10.1016/0167-2789(94)00220-K.  Google Scholar [42] M. J. Raković and Shih-I Chu, Phase-space structure of a new integrable system related to hydrogen atoms in external fields, J. Phys. A, 30 (1997), 733-753. doi: 10.1088/0305-4470/30/2/033.  Google Scholar [43] H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, in "Dynamical Systems, Theory and Applications" (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Phys., Vol. 38, Berlin, Springer, (1975), 598-624.  Google Scholar [44] C. Simó, private communication private communication, 2000. Google Scholar [45] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. II, Comm. Pure Appl. Math., 29 (1976), 49-111. doi: 10.1002/cpa.3160290104.  Google Scholar
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