Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Low dimensional completely resonant tori in Hamiltonian Lattices and a Theorem of Poincaré

1 Department of Mathematics “F. Enriques”, Milano University, via Saldini 50, 20133 - Milano, Italy
2 GNFM (Gruppo Nazionale di Fisica Matematica) – Indam (Istituto Nazionale di Alta Matematica “F. Severi”), Roma, Italy

This contribution is part of the Special Issue: Modern methods in Hamiltonian perturbation theory
Guest Editors: Marco Sansottera; Ugo Locatelli
Link: www.aimspress.com/mine/article/5514/special-articles

Special Issues: Modern methods in Hamiltonian perturbation theory

We present an extension of a classical result of Poincaré (1892) about continuation of periodic orbits and breaking of completely resonant tori in a class of nearly integrable Hamiltonian systems, which covers most Hamiltonian Lattice models. The result is based on the fixed point method of the period map and exploits a standard perturbation expansion of the solution with respect to a small parameter. Two different statements are given, about existence and linear stability: a first one, in the so called non-degenerate case, and a second one, in the completely degenerate case. A pair of examples inspired to the existence of localized solutions in the discrete NLS lattice is provided.
  Figure/Table
  Supplementary
  Article Metrics

Keywords Hamiltonian lattices; perturbation theory; average methods; resonant tori; periodic orbits; linear stability

Citation: Tiziano Penati, Veronica Danesi, Simone Paleari. Low dimensional completely resonant tori in Hamiltonian Lattices and a Theorem of Poincaré. Mathematics in Engineering, 2021, 3(4): 1-20. doi: 10.3934/mine.2021029

References

  • 1. Kapitula T, Kevrekidis P (2001) Stability of waves in discrete systems. Nonlinearity 14: 533-566.    
  • 2. Ahn T, Mackay RS, Sepulchre JA (2001) Dynamics of relative phases: Generalised multibreathers. Nonlinear Dynam 25: 157-182.    
  • 3. Aubry S (1997) Breathers in nonlinear lattices: Existence, linear stability and quantization. Phys D 103: 201-250.    
  • 4. Bruno AD (2020) Normalization of a periodic Hamiltonian system. Program Comput Soft 46: 76- 83.    
  • 5. Cheng CQ, Wang S (1999) The surviving of lower dimensional tori from a resonant torus of Hamiltonian Systems. J Differ Equations 155: 311-326.    
  • 6. Cuevas J, Koukouloyannis V, Kevrekidis PG, et al. (2011) Multibreather and vortex breather stability in Klein-Gordon lattices: Equivalence between two different approaches. Int J Bifurcat Chaos 21: 2161-2177.    
  • 7. Ekeland I (1990) Convexity Methods in Hamiltonian Mechanics, Berlin: Springer-Verlag.
  • 8. Graff SM (1974) On the conservation of hyperbolic invariant for Hamiltonian systems. J Differ Equations 15: 1-69.    
  • 9. Han Y, Li Y, Yi Y (2006) Degenerate lower dimensional tori in Hamiltonian Systems. J Differ Equations 227: 670-691.    
  • 10. Kapitula T (2001) Stability of waves in perturbed Hamiltonian systems. Phys D 156: 186-200.    
  • 11. Koukouloyannis V, Kevrekidis PG (2009) On the stability of multibreathers in Klein-Gordon chains. Nonlinearity 22: 2269-2285.    
  • 12. Kevrekidis PG (2009) The Discrete Nonlinear Schrödinger Equation, Berlin: Springer-Verlag.
  • 13. Kevrekidis PG (2009) Non-nearest-neighbor interactions in nonlinear dynamical lattices. Phys Lett A 373: 3688-3693.    
  • 14. Koukouloyannis V, Kevrekidis PG, Cuevas J, et al. (2013) Multibreathers in Klein-Gordon chains with interactions beyond nearest neighbors. Phys D 242: 16-29.    
  • 15. Koukouloyannis V (2013) Non-existence of phase-shift breathers in one-dimensional KleinGordon lattices with nearest-neighbor interactions. Phys Lett A 377: 2022-2026.    
  • 16. Li Y, Yi Y (2003) A quasi-periodic Poincaré's theorem. Math Ann 326: 649-690.    
  • 17. MacKay RS (1996) Dynamics of networks: Features which persist from the uncoupled limit, In: Stochastic and Spatial Structures of Dynamical Systems, Amsterdam: North-Holland, 81-104.
  • 18. MacKay RS, Sepulchre JS (1998) Stability of discrete breathers. Phys D 119: 148-162.    
  • 19. Meletlidou E, Ichtiaroglou S (1994) On the number of isolating integrals in perturbed Hamiltonian systems with n ≥ 3 degrees of freedom. J Phys A Math Gen 27: 3919-3926.    
  • 20. Meletlidou E, Stagika G (2006) On the continuation of degenerate periodic orbits in Hamiltonian systems. Regul Chaotic Dyn 11: 131-138.    
  • 21. Paleari S, Penati T (2019) Hamiltonian lattice dynamics. Mathematics in Engineering 1: 881-887.    
  • 22. Pelinovsky DE, Kevrekidis PG, Frantzeskakis DJ (2005) Persistence and stability of discrete vortices in nonlinear Schrödinger lattices. Phys D 212: 20-53.    
  • 23. Pelinovsky DE, Kevrekidis PG, Frantzeskakis DJ (2005) Stability of discrete solitons in nonlinear Schrödinger lattices. Phys D 212: 1-19.    
  • 24. Penati T, Sansottera M, Paleari S, et al. (2018) On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice. Phys D 370: 1-13.    
  • 25. Penati T, Koukouloyannis V, Sansottera M, et al. (2019) On the nonexistence of degenerate phaseshift multibreathers in Klein-Gordon models with interactions beyond nearest neighbors. Phys D 398: 92-114.    
  • 26. Penati T, Sansottera M, Danesi V (2018) On the continuation of degenerate periodic orbits via normal form: Full dimensional resonant tori. Commun Nonlinear Sci 61: 198-224.    
  • 27. Sansottera M, Danesi V, Penati T, et al. (2020) On the continuation of degenerate periodic orbits via normal form: Lower dimensional resonant tori. Commun Nonlinear Sci 90: 105360.    
  • 28. Poincaré H (1957) Les méthodes nouvelles de la mécanique céleste. Tome I. Solutions périodiques. Non-existence des intégrales uniformes. Solutions asymptotiques. New York: Dover Publications, Inc.
  • 29. Poincaré H (1996) OEuvres. Tome VII. Les Grands Classiques Gauthier-Villars, Sceaux: Jacques Gabay.
  • 30. Treshchev DV (1991) The mechanism of destruction of resonant tori of Hamiltonian systems. Math USSR Sb 68: 181-203.    
  • 31. Voyatzis G, Ichtiaroglou S (1999) Degenerate bifurcations of resonant tori in Hamiltonian systems. Int J Bifurcat Chaos 9: 849-863.    
  • 32. Yakubovich VA, Starzhinskii VM (1975) Linear differential equations with periodic coefficients. 1 and 2. New York-Toronto: John Wiley & Sons.

 

Reader Comments

your name: *   your email: *  

© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved