Search Advanced

Mathematics in Engineering

2021, Volume 3, Issue 4: 1-20. doi: 10.3934/mine.2021029
Previous Article Next Article
Research article Special Issues

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
  • Received: 20 February 2020 Accepted: 31 May 2020 Published: 03 August 2020

  • 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.

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

    Related Papers:

  • 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.


    加载中


    [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 Schr?dinger 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 Schr?dinger lattices. Phys D 212: 20-53.
    [23] Pelinovsky DE, Kevrekidis PG, Frantzeskakis DJ (2005) Stability of discrete solitons in nonlinear Schr?dinger 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
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematics in Engineering
1.4 2.2

Metrics

Article views(3767) PDF downloads(675) Cited by(2)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog