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Parameterization Method for Computing Quasi-periodic Reducible Normally Hyperbolic Invariant Tori

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Advances in Differential Equations and Applications

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 4))

Abstract

We consider the problem of numerically computing quasi-periodic normally hyperbolic invariant tori (NHIT) with fixed frequency as well as their invariant bundles. The algorithm is based on a KAM scheme to find the parameterization of a torus with fixed Diophantine frequency (by adjusting parameters of the model), and suitable Floquet transformations that reduce the linearized dynamics to constant coefficients. We apply this method to continue curves of quasi-periodic NHIT of a perturbed dynamical system and to explore the mechanism of breakdown of these invariant tori. We observe in these continuations that the invariant bundles may collide even if the Lyapunov multipliers remain separated.

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References

  1. Broer, H.W., Huitema, G.B., Sevryuk, M.B.: Quasi-periodic Motions in Families of Dynamical Systems. Order Amidst Chaos. Lecture Notes in Mathematics, vol. 1645. Springer, Berlin (1996)

    Google Scholar 

  2. Broer, H., Osinga, H., Vegter, G.: Algorithms for computing normally hyperbolic invariant manifolds. Z. Angew. Math. Phys. 48(3), 480–524 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  5. Canadell, M.: Computation of normally hyperbolic invariant manifolds. PhD thesis, Universitat de Barcelona (July 2014)

    Google Scholar 

  6. Canadell, M., Haro, A.: A Newton-like method for computing normally hyperbolic invariant tori. Chapter 5 of the parameterization method for invariant manifolds: from rigorous results to effective computations (2014, in progress)

    Google Scholar 

  7. Canadell, M., Haro, A.: A KAM-like theorem for Quasi-Periodic Normally Hyperbolic Invariant Tori (2014, in progress)

    Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Figueras, J.L.: Fiberwise hyperbolic invariant Tori in quasiperiodically skew product systems. PhD. thesis, Universitat de Barcelona (2011)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Haro, A., de la Llave, R.: A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: numerical algorithms. Discret. Contin. Dyn-B. 6, 1261–1300 (2006)

    Article  MATH  Google Scholar 

  13. 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  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  15. Osinga, H., Schilder, F., Vogt, W.: Continuation of quasi-periodic invariant tori. SIAM J. Appl. Dyn. Syst. 4(3), 459–488 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

M.C. and A.H. have been funded by the Spanish grants MTM2009-09723 and MTM2012-32541. M.C. has also been funded by the FPI grant BES-2010-039663 and A.H. by the Catalan grant 2009-SGR-67.

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Authors and Affiliations

  1. Departament de Matemática Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    Marta Canadell & Àlex Haro

  2. School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, 30332, Atlanta, Georgia, USA

    Marta Canadell & Àlex Haro

Authors
  1. Marta Canadell
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  2. Àlex Haro
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Editors and Affiliations

  1. Dept. de Mathemàtiques and IMAC, Universitat Jaume I, Castelló, Spain

    Fernando Casas

  2. Dept. de Mathemàtiques and IMAC, Universitat Jaume I, Castelló, Spain

    Vicente Martínez

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Canadell, M., Haro, À. (2014). Parameterization Method for Computing Quasi-periodic Reducible Normally Hyperbolic Invariant Tori. In: Casas, F., Martínez, V. (eds) Advances in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-06953-1_9

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