Abstract
In this note, we describe the setting and main results concerning the existence of invariant tori and a class of invariant manifolds for differentiable skew product systems in lattices having interactions with spatial decay among all nodes. We obtain decay of the parameterizations of the objects we find.
This work has been supported by grants MTM2016-80117-P, MDM2014-0445 (Spain) and 2017-SGR-1374 (Catalonia).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
D. Blazevski, R. de la Llave, Localized stable manifolds for whiskered tori in coupled map lattices with decaying interaction. Ann. Henri Poincaré 15(1), 29–60 (2014)
O.M. Braun, Y.S. Kivshar, The Frenkel–Kontorova model: concepts, methods, and applications. Texts and Monographs in Physics (Springer, Berlin, 2004)
X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds, I. Manifolds associated to non-resonant subspaces. Indiana Univ. Math. J. 52(2), 283–328 (2003)
X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds, II. Regularity with respect to parameters. Indiana Univ. Math. J. 52(2), 329–360 (2003)
U. Dehlinger, Zur theorie der rekristallisation reiner metalle. Annalen der Physik 394(7), 749–793 (1929)
E. Fontich, R. de la Llave, P. Martín, Dynamical systems on lattices with decaying interaction I: a functional analysis framework. J. Differ. Equ. 250(6), 2838–2886 (2011)
E. Fontich, R. de la Llave, Y. Sire, Construction of invariant whiskered tori by a parameterization method, I. Maps and flows in finite dimensions. J. Differ. Equ. 246(8), 3136–3213 (2009)
Y.I. Frenkel, T.A. Kontorova, The model of dislocation in solid body. Zh. Eksp. Teor. Fiz 8 (1938)
Y.I. Frenkel, T.A. Kontorova, On the theory of plastic deformation and twinning. Acad. Sci. U.S.S.R. J. Phys. 1, 137–149 (1939)
A. Haro, M. Canadell, J.L. Figueras, A. Luque, J.M. Mondelo, The parameterization method for invariant manifolds. From rigorous results to effective computations. Appl. Math. Sci. 195 (2016). Springer
A. Haro, R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results. J. Differ. Equ. 228(2), 530–579 (2006)
M. Jiang, R. de la Llave, Smooth dependence of thermodynamic limits of SRB-measures. Comm. Math. Phys. 211(2), 303–333 (2000)
R. de la Llave, A. González, À. Jorba, J. Villanueva, KAM theory without action-angle variables. Nonlinearity 18(2), 855–895 (2005)
L. Prandtl, A conceptual model to the kinetic theory of solid bodies. Z. Angew. Math. Mech. 8, 85–106 (1928)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Berenguel, R., Fontich, E. (2019). Invariant Objects on Lattice Systems with Decaying Interactions. In: Korobeinikov, A., Caubergh, M., Lázaro, T., Sardanyés, J. (eds) Extended Abstracts Spring 2018. Trends in Mathematics(), vol 11. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-25261-8_21
Download citation
DOI: https://doi.org/10.1007/978-3-030-25261-8_21
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-25260-1
Online ISBN: 978-3-030-25261-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)