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Dynamics of a Chain with Four Particles, Alternating Masses and Nearest-Neighbor Interaction

Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 199)

Abstract

We formulate the periodic FPU problem with four alternating masses which is the simplest nontrivial version. The analysis involves normal form calculations to second order producing integrable normal forms with three timescales. In the case of large alternating mass the system is an example of dynamics with widely separated frequencies and three timescales. The presence of approximate integrals and the stability characteristics of the periodic solutions lead to weak interaction of the modes of the system.

References

  1. 1.
    Bruggeman, R., Verhulst, F.: The inhomogenous Fermi-Pasta-Ulam chain, submitted (2015). arXiv:1510.00560 [math.DS]
  2. 2.
    Galgani, L., Giorgilli, A., Martinoli, A., Vanzini, S.: On the problem of energy partition for large systems of the Fermi-Pasta-Ulam type: analytical and numerical estimates. Phys. D 59, 334–348 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Poincaré, Henri: Les Méthodes Nouvelles de la Mécanique Célèste, 3 vols. Gauthier-Villars, Paris 1892, 1893 (1899)ADSGoogle Scholar
  4. 4.
    Rink, B., Verhulst, F.: Near-integrability of periodic FPU-chains. Phys. A 285, 467–482 (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Rink, B.: Symmetry and resonance in periodic FPU-chains. Comm. Math. Phys. 218, 665–685 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Sanders, J.A., Verhulst, F., Murdock, J.: Averaging methods in nonlinear dynamical systems. Appl. Math. Sci. vol. 59, 2nd ed., Springer (2007)Google Scholar
  7. 7.
    Tuwankotta, J.M., Verhulst, F.: Hamiltonian systems with widely separated frequencies. Nonlinearity 16, 689–706 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Verhulst, F.: Methods and Applications of Singular Perturbations. Springer, Berlin (2005)Google Scholar
  9. 9.
    Verhulst, F.: A chain of FPU cells. In: Awrejcewicz, J., Kazmierczak, M., Mrozowski, J., Oleinik, P. (eds.) In: Proceedings of the 13th International Conference Dynamical Systems—Theory and Applications, vol. Control and Stability, pp. 603–612, Lodz, December 7–10 (2015). Extended version. Regular and chaotic recurrence in FPU cell-chains publ. in Appl. Math. Modelling (2016)Google Scholar
  10. 10.
    Verhulst, F.: Near-integrability and recurrence in FPU cells. Int. J. Bif. Chaos, accepted for publ. (2017)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Mathematisch InstituutUtrechtNetherlands

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