American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 323-333. doi: 10.3934/proc.2013.2013.323

Fast iteration of cocycles over rotations and computation of hyperbolic bundles

Gemma Huguet 1, , Rafael de la Llave 2, and  Yannick Sire 3,

1. 

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, United States
2. 

School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA 30332-0160, United States
3. 

Université Paul Cézanne, Laboratoire LATP UMR 6632, Marseille

Received  July 2012 Revised  February 2013 Published  November 2013

We present numerical algorithms that use small requirements of storage and operations to compute the iteration of cocycles over a rotation. We also show that these algorithms can be used to compute efficiently the stable and unstable bundles and the Lyapunov exponents of the cocycle.
Keywords: quasi-periodic cocycles, Quasi-periodic solutions, numerical computation..
Mathematics Subject Classification: Primary: 70K43; Secondary: 37J4.
Citation: Gemma Huguet, Rafael de la Llave, Yannick Sire. Fast iteration of cocycles over rotations and computation of hyperbolic bundles. Conference Publications, 2013, 2013 (special) : 323-333. doi: 10.3934/proc.2013.2013.323
References:
[1]

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, "LAPACK user's guide'', 3rd edition, Software - Environments - Tools. 9. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics, 1999.

[2]

J. Bourgain, "Green's function estimates for lattice Schrödinger operators and applications'', volume 158 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2005.

[3]

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.

[4]

L. Dieci and E. S. Van Vleck, Lyapunov spectral intervals: theory and computation, SIAM J. Numer. Anal., 40 (2002), 516-542.

[5]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656.

[6]

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

[7]

G. H. Golub and C. F. Van Loan, "Matrix computations'', Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, third edition, 1996.

[8]

À. Haro and R. d. l. Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8 pp.

[9]

À. Haro and R. d. l. 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.

[10]

A. Haro and R. d. l. 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.

[11]

A. Haro and R. de la Llave, A parameterization method for the computation of whiskers in quasi periodic maps: numerical implementation and examples, SIAM Jour. Appl. Dyn. Syst., 6 (2007), 142-207.

[12]

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), 1309-1353.

[13]

R. Krikorian, $C^0$-densité globale des systèmes produits-croisés sur le cercle r\'eductibles, Ergodic Theory Dynam. Systems, 19 (1999), 61-100.

[14]

R. Krikorian, "Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts'', Astérisque, (259):vi+216, 1999.

[15]

K. R. Meyer and G. R. Sell, Melnikov transforms, Bernoulli bundles, and almost periodic perturbations, Trans. Amer. Math. Soc., 314 (1989), 63-105.

[16]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.

[17]

L. Pastur and A. Figotin, "Spectra of random and almost-periodic operators'', volume 297 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1992.

[18]

J. Puig, Reducibility of linear differential equations with quasi-periodic coefficients: a survey, preprint, mp_arc/02-246.

[19]

M. Rychlik, Renormalization of cocycles and linear ODE with almost-periodic coefficients, Invent. Math., 110 (1992), 173-206.

[20]

R. Sacker, Existence of dichotomies and invariant splittings for linear differential systems. IV, J. Differential Equations, 27 (1978), 106-137.

[21]

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

[22]

R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. II, J. Differential Equations, 22 (1976), 478-496.

[23]

R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. III, J. Differential Equations, 22 (1976), 497-522.

show all references

References:
[1]

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, "LAPACK user's guide'', 3rd edition, Software - Environments - Tools. 9. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics, 1999.

[2]

J. Bourgain, "Green's function estimates for lattice Schrödinger operators and applications'', volume 158 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2005.

[3]

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.

[4]

L. Dieci and E. S. Van Vleck, Lyapunov spectral intervals: theory and computation, SIAM J. Numer. Anal., 40 (2002), 516-542.

[5]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656.

[6]

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

[7]

G. H. Golub and C. F. Van Loan, "Matrix computations'', Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, third edition, 1996.

[8]

À. Haro and R. d. l. Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8 pp.

[9]

À. Haro and R. d. l. 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.

[10]

A. Haro and R. d. l. 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.

[11]

A. Haro and R. de la Llave, A parameterization method for the computation of whiskers in quasi periodic maps: numerical implementation and examples, SIAM Jour. Appl. Dyn. Syst., 6 (2007), 142-207.

[12]

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), 1309-1353.

[13]

R. Krikorian, $C^0$-densité globale des systèmes produits-croisés sur le cercle r\'eductibles, Ergodic Theory Dynam. Systems, 19 (1999), 61-100.

[14]

R. Krikorian, "Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts'', Astérisque, (259):vi+216, 1999.

[15]

K. R. Meyer and G. R. Sell, Melnikov transforms, Bernoulli bundles, and almost periodic perturbations, Trans. Amer. Math. Soc., 314 (1989), 63-105.

[16]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.

[17]

L. Pastur and A. Figotin, "Spectra of random and almost-periodic operators'', volume 297 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1992.

[18]

J. Puig, Reducibility of linear differential equations with quasi-periodic coefficients: a survey, preprint, mp_arc/02-246.

[19]

M. Rychlik, Renormalization of cocycles and linear ODE with almost-periodic coefficients, Invent. Math., 110 (1992), 173-206.

[20]

R. Sacker, Existence of dichotomies and invariant splittings for linear differential systems. IV, J. Differential Equations, 27 (1978), 106-137.

[21]

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

[22]

R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. II, J. Differential Equations, 22 (1976), 478-496.

[23]

R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. III, J. Differential Equations, 22 (1976), 497-522.

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