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Near-Integrability and Recurrence in FPU-Cells

Ferdinand Verhulst

Mathematisch Instituut, P. O. Box 80.010, 3508TA Utrecht, Netherlands

https://doi.org/10.1142/S0218127416502308Cited by:8 (Source: Crossref)

Abstract

In a neighborhood of stable equilibrium, we consider the dynamics for at least three degrees-of-freedom (dof) Hamiltonian systems (2 dof systems are not ergodic in this case). A complication is that the recurrence properties depend strongly on the resonances of the corresponding linearized system and on quasi-trapping. In contrast to the classical FPU-chain, the inhomogeneous FPU-chain shows nearly all the principal resonances. Using this fact, we construct a periodic FPU-chain of low dimension, called a FPU-cell. Such a cell can be used as a building block for a chain of FPU-cells, called a cell-chain. Recurrence phenomena depend strongly on the physical assumptions producing specific Hamiltonians; we demonstrate this for the $1 : 2 : 5$ $1 : 2 : 5$ resonance, both general and for the FPU case; this resonance shows dynamics on different timescales. In addition we will study the relations and recurrence differences between several FPU-cells and a few cell-chains in the case of the classical near-integrable FPU-cell and of chaotic cells in $3 : 2 : 1$ $3 : 2 : 1$ resonance.

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References

Bruggeman, R. & Verhulst, F. [2015] “The Inhomogenous Fermi–Pasta–Ulam chain,” submitted, arXiv:1510. 00560 [math.DS]. Google Scholar
Chechin, G. M. & Sakhnenko, V. P. [1998] “ Interaction between normal modes in nonlinear dynamical systems with discrete symmetry. Exact results,” Physica D 117, 43–76. Crossref, Web of Science, Google Scholar
Chechin, G. M., Novikova, N. V. & Abramenko, A. A. [2002] “ Bushes of vibrational normal modes for Fermi–Pasta–Ulam chains,” Physica D 166, 208–238. Crossref, Web of Science, Google Scholar
Christodoulidi, H., Efthymiopoulos, Ch. & Bountis, T. [2010] “ Energy localization on $q$ $q$ -tori, long-term stability, and the interpretation of Fermi–Pasta–Ulam recurrences,” Phys. Rev. E 81, 6210. Crossref, Web of Science, Google Scholar
Contopoulos, G. & Polymilis, C. [1996] “ Recurrence in the homoclinic tangle,” Celest. Mech. Dyn. Astron. 63, 189–197. Crossref, Web of Science, Google Scholar
Devaney, R. L. [1976] “ Homoclinic orbits in Hamiltonian systems,” J. Diff. Eqs. 21, 431–438. Crossref, Web of Science, Google Scholar
Fermi, E., Pasta, J. & Ulam, S. [1955] “Los Alamos report LA-1940,” in E. Fermi, Collected Papers, Vol. 2, pp. 977–988. Google Scholar
Ford, J. [1992] “ The Fermi–Pasta–Ulam problem: Paradox turns discovery,” Phys. Rep. 213, 271–310. Crossref, Web of Science, Google Scholar
Hairer, E., Lubich, C. & Wanner, G. [2006] Geometric Numerical Integration, 2nd edition (Springer). Google Scholar
Haller, G. & Wiggins, S. [1996] “ Geometry and chaos near resonant equilibria of $3$ $3$ -DOF Hamiltonian systems,” Physica D 90, 319–365. Crossref, Web of Science, Google Scholar
Hoveijn, I. & Verhulst, F. [1990] “ Chaos in the $1 : 2 : 3$ $1 : 2 : 3$ Hamiltonian normal form,” Physica D 44, 397–406. Crossref, Web of Science, Google Scholar
Poggi, P. & Ruffo, S. [1997] “ Exact solutions in the FPU oscillator chain,” Physica D 103, 251–272. Crossref, Web of Science, Google Scholar
Poincaré, H. [1892, 1893, 1899] Les Méthodes Nouvelles de la Mécanique Célèste, Vol. 3 (Gauthier-Villars, Paris). Google Scholar
Rink, B. & Verhulst, F. [2000] “ Near-integrability of periodic FPU-chains,” Physica A 285, 467–482. Crossref, Web of Science, Google Scholar
Rink, B. [2001] “ Symmetry and resonance in periodic FPU-chains,” Commun. Math. Phys. 218, 665–685. Crossref, Web of Science, Google Scholar
Sanders, J. A., Verhulst, F. & Murdock, J. [2007] Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, Vol. 59, 2nd edition (Springer). Google Scholar
Van der Aa, E. & De Winkel, M. [1994] “ Hamiltonian systems in $1 : 2 : ω$ $1 : 2 : ω$ -resonance,” Int. J. Non-Lin. Mech. 29, 261–270. Crossref, Web of Science, Google Scholar
Verhulst, F. [2015a] “ Integrability and non-integrability of Hamiltonian normal forms,” Acta Appl. Math. 137, 253–272. Crossref, Web of Science, Google Scholar
Verhulst, F. [2015b] “ A chain of FPU-cells,” Proc. 13th Int. Conf. Dynamical Systems — Theory and Applications, eds. Awrejcewicz, J., Kazmierczak, M., Mrozowski, J. & Oleinik, P., Lodz, December 7–10, pp. 603–612. Extended version in Applied Mathematical Modelling, http://dx.doi.org/10.1016/ j.apm.2016.06.047. Google Scholar
Zaslavsky, G. M. [2007] The Physics of Chaos in Hamiltonian Systems, 2nd extended edition (Imperial College Press). Link, Google Scholar