[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

Computation of Quasiperiodic Normally Hyperbolic Invariant Tori: Rigorous Results

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

The development of efficient methods for detecting quasiperiodic oscillations and computing the corresponding invariant tori is a subject of great importance in dynamical systems and their applications in science and engineering. In this paper, we prove the convergence of a new Newton-like method for computing quasiperiodic normally hyperbolic invariant tori carrying quasiperiodic motion in smooth families of real-analytic dynamical systems. The main result is stated as an a posteriori KAM-like theorem that allows controlling the inner dynamics on the torus with appropriate detuning parameters, in order to obtain a prescribed quasiperiodic motion. The Newton-like method leads to several fast and efficient computational algorithms, which are discussed and tested in a companion paper (Canadell and Haro in J Nonlinear Sci, 2017. doi:10.1007/s00332-017-9388-z), in which new mechanisms of breakdown are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Adomaitis, R.A., Kevrekidis, I.G., de la Llave, R.: Predicting the complexity of disconnected basins of attraction for a noninvertible system. Technical Report, Systems Research Center Technical Report TR91-41 (1991)

  • Adomaitis, R.A., Kevrekidis, I.G., de la Llave, R.: A computer-assisted study of global dynamic transitions for a noninvertible system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 17(4), 1305–1321 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Broer, H.W., Huitema, G.B., Sevryuk, M.B.: Quasi-periodic motions in families of dynamical systems. In: Order Amidst Chaos. Lecture Notes in Math., vol. 1645. Springer, Berlin (1996)

  • Bourgain, J.: On Melnikov’s persistency problem. Math. Res. Lett. 4(4), 445–458 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Calleja, R., Celletti, A., de la Llave, R.: A KAM theory for conformally symplectic systems: efficient algorithms and their validation. J. Differ. Equ. 255(5), 978–1049 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Calleja, R., de la Llave, R.: A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification. Nonlinearity 23(9), 2029–2058 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Calleja, R., Figueras, J.L.: Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map. Chaos 22(3), 033114 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Cabré, X., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces. Indiana Univ. Math. J. 52(2), 283–328 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Canadell, M., Haro, A.: Computation of quasi-periodic normally hyperbolic invariant tori: algorithms, numerical explorations and mechanisms of breakdown. J. Nonlinear Sci. (2017). doi:10.1007/s00332-017-9388-z

  • Chan, T.N.: Numerical bifurcation analysis of simple dynamical systems. Ph.D. thesis, Concordia University, Montreal, Canada, September (1983)

  • Castellà, E., Jorba, À.: On the vertical families of two-dimensional tori near the triangular points of the bicircular problem. Celest. Mech. Dyn. Astrono. 76(1), 35–54 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Díez, C., Jorba, À., Simó, C.: A dynamical equivalent to the equilateral libration points of the real Earth–Moon system. Celest. Mech. 50(1), 13–29 (1991)

    Article  MATH  Google Scholar 

  • de la Llave, R.: Hyperbolic dynamical systems and generation of magnetic fields by perfectly conducting fluids. Geophys. Astrophys. Fluid Dyn. 73(1–4), 123–131 (1993)

    Article  MathSciNet  Google Scholar 

  • de la Llave, R.: A tutorial on KAM theory, Smooth ergodic theoryand its applications (Seattle, WA, 1999). In: Proceedings of Symposium in Pure Mathematics, vol. 69, pp. 175–292. Amer. Math. Soc., Providence, RI (2001)

  • de la Llave, R., González, A., Jorba, À., Villanueva, J.: KAM theory without action-angle variables. Nonlinearity 18(2), 855–895 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Fontich, E., de la Llave, R., Sire, Y.: Construction of invariant whiskered tori by a parameterization method. I. Maps and flows in finite dimensions. J. Differ. Equ. 246(8), 3136–3213 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

  • Figueras, J.-Ll., Haro, A., Luque, A.: Rigorous computer assisted application of KAM theory: a modern approach. Found. Comput. Math. (2016). doi:10.1007/s10208-016-9339-3

  • Franceschini, V., Russo, L.: Stable and unstable manifolds of the Hénon mapping. J. Stat. Phys. 25(4), 757–769 (1981)

    Article  MATH  Google Scholar 

  • González, A., de la Llave, R., Haro, A.: An introduction to singularity theory for non-twist KAM tori (in progress)

  • González, A., Haro, A., de la Llave, R.: Singularity theory for non-twist KAM tori. Mem. Am. Math. Soc. 227(1067), vi+115 (2014)

    MathSciNet  MATH  Google Scholar 

  • Haro, A.: An algorithm to generate canonical transformations: application to normal forms. Phys. D 167(3–4), 197–217 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Haro, A., Canadell, M., Figueras, J-LL., Luque, A., Mondelo, J-M.: The parameterization method for invariant manifolds. Appl. Math. Sci., vol. 195, Springer. http://www.springer.com/us/book/9783319296609 (2016)

  • Haro, A., de la Llave, R.: Spectral theory and dynamical systems (2005). http://www.maia.ub.es/~alex/spectrum-dynamics/spectrum-dynamics.pdf

  • Haro, A., de la Llave, R.: Manifolds on the verge of a hyperbolicity breakdown. Chaos 16(1), 013120 (2006a)

    Article  MathSciNet  MATH  Google Scholar 

  • Haro, A., de la Llave, R.: 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(6), 1261–1300 (2006b)

    Article  MathSciNet  MATH  Google Scholar 

  • Haro, A., de la Llave, R.: A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results. J. Differ. Equ. 228(2), 530–579 (2006c)

    Article  MathSciNet  MATH  Google Scholar 

  • Haro, A., de la Llave, R.: 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(1), 142–207 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Huguet, G., de la Llave, R., Sire, Y.: Computation of whiskered invariant tori and their associated manifolds: new fast algorithms. Discrete Contin. Dyn. Syst. 32(4), 1309–1353 (2012)

    MathSciNet  MATH  Google Scholar 

  • Herman, M.-R.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol’ d et de Moser sur le tore de dimension \(2\). Comment. Math. Helv. 58(3), 453–502 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. Lecture Notes in Mathematics, vol. 583. Springer, Berlin (1977)

  • Jorba, À., Ollé, M.: Invariant curves near Hamiltonian–Hopf bifurcations of four-dimensional symplectic maps. Nonlinearity 17(2), 691–710 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Johnson, R.A.: Analyticity of spectral subbundles. J. Differ. Equ. 35(3), 366–387 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  • Jorba, À.: Numerical computation of the normal behaviour of invariant curves of \(n\)-dimensional maps. Nonlinearity 14(5), 943–976 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Kato, T.: Perturbation theory for linear operators. In: Classics in Mathematics. Springer, Berlin (1995) (reprint of the 1980 edition)

  • Latushkin, YuD, Stëpin, A.M.: Weighted shift operators, the spectral theory of linear extensions and a multiplicative ergodic theorem. Math. Sb. 181(6), 723–742 (1990)

    MathSciNet  Google Scholar 

  • Luque, A., Villanueva, J.: A KAM theorem without action-angle variables for elliptic lower dimensional tori. Nonlinearity 24(4), 1033–1080 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Mañé, R.: Persistent manifolds are normally hyperbolic. Trans. Am. Math. Soc. 246, 261–283 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  • Mather, J.N.: Characterization of Anosov diffeomorphisms. Indag. Math. 30, 479–483 (1968)

    Article  MathSciNet  MATH  Google Scholar 

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

  • Mondelo, J.M., Barrabés, E., Gómez, G., Ollé, M.: 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)

  • Moser, J.: A rapidly convergent iteration method and non-linear partial differential equations. I. Ann. Sc. Norm. Super. Pisa 20, 265–315 (1966a)

    MathSciNet  MATH  Google Scholar 

  • Moser, J.: A rapidly convergent iteration method and non-linear differential equations. II. Ann. Sc. Norm. Super. Pisa 20, 499–535 (1966b)

    MathSciNet  MATH  Google Scholar 

  • Moser, J.: Convergent series expansions for quasi-periodic motions. Math. Ann. 169, 136–176 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  • Peckham, B.B., Schilder, F.: Computing Arnol’d tongue scenarios. J. Comput. Phys. 220(2), 932–951 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Rüssmann, H.: On optimal estimates for the solutions of linear difference equations on the circle. In: Proceedings of the Fifth Conference on Mathematical Methods in Celestial Mechanics (Oberwolfach, 1975), Part I. Celestial Mech., vol. 14 (1976)

  • Simó, C.: On the Analytical and numerical approximation of invariant manifolds, modern methods in celestial mechanics. In: Comptes Rendus de la 13ieme Ecole Printemps d’Astrophysique de Goutelas (France), 24–29 Avril, 1989. Edited by Daniel Benest and Claude Froeschlé, p. 285. Gif-sur-Yvette: Editions Frontieres (1990)

  • Sacker, R.J., Sell, G.R.: Existence of dichotomies and invariant splittings for linear differential systems. I. J. Differ. Equ. 15, 429–458 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  • Takens, F., Wagener, F.O.O.: Resonances in skew and reducible quasi-periodic Hopf bifurcations. Nonlinearity 13(2), 377–396 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Vitolo, R., Broer, H., Simó, C.: Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems. Regul. Chaotic Dyn. 16(1–2), 154–184 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Wagener, F.O.O.: On the skew hopf bifurcation. Ph.D. thesis, University of Groningen (1998)

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

    Article  MathSciNet  MATH  Google Scholar 

  • Zehnder, E.: Generalized implicit function theorems with applications to some small divisor problems. II. Commun. Pure Appl. Math. 29(1), 49–111 (1976)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Rafael de la Llave, Alejandro Luque, and Carles Simó for fruitful discussions and comments. We also thank the referees for their insightful comments and suggestions on the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Àlex Haro.

Additional information

Communicated by Eusebius Doedel.

Marta Canadell and Àlex Haro acknowledge support from the Spanish Grants MTM2012-32541 and MTM2015-67724-P, and the Catalan Grant 2014-SGR-1145. Marta Canadell also acknowledges support from the FPI Grant BES-2010-039663, and the NSF Grant DMS-1500943.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Canadell, M., Haro, À. Computation of Quasiperiodic Normally Hyperbolic Invariant Tori: Rigorous Results. J Nonlinear Sci 27, 1869–1904 (2017). https://doi.org/10.1007/s00332-017-9389-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-017-9389-y

Keywords

Mathematics Subject Classification

Navigation