Perturbation theory for periodic orbits in a class of infinite dimensional Hamiltonian systems

Albanese, Claudio; Fröhlich, Jürg

doi:10.1007/bf02099674

Cited by 36 publications

(40 citation statements)

References 6 publications

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“…Another intriguing question is the relation between time-periodic and spatially localized solutions in nonlinear disordered lattices [7,366,219,14,221,220,197] and the diffusion of an initially localized excitation [367,283]. This is currently an active field of research, and we can expect to receive fresh news in the very near future.…”

Section: Q-breathers Localization In Normal Mode Space and The Fermmentioning

confidence: 97%

“…Typical propagation distances in waveguide arrays are rather short, so that usually the effects of light dispersion in each individual waveguide are neglected. 7 Under this assumption, the total field distribution in a one-dimensional waveguide array, see Fig. 63, can be approximated as the superposition of linear waveguide modes of each individual waveguide E(x, z) = n E n (z)ψ n (x) exp(−iλ 0 z), (10.9) where ψ n (x) ≡ ψ 0 (x − nd), ψ 0 (x) is the linear waveguide mode with corresponding propagation constant λ 0 :…”

Section: Basic Principles and Modelingmentioning

confidence: 97%

See 1 more Smart Citation

Discrete breathers — Advances in theory and applications

Flach¹,

Gorbach²

2008

Physics Reports

940

830

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Section: Q-breathers Localization In Normal Mode Space and The Fermmentioning

confidence: 97%

Section: Basic Principles and Modelingmentioning

confidence: 97%

Discrete breathers — Advances in theory and applications

Flach¹,

Gorbach²

2008

Physics Reports

940

830

View full text Add to dashboard Cite

“…¿From a mathematical point of view, Albanese and Fröhlich have proved the existence of breathers for a class of random Hamiltonian systems describing an infinite array of coupled anharmonic oscillators [AF91] (see also the earlier work [FSW86] of Fröhlich et al concerning quasiperiodic localized oscillations). These breather families can be parametrized by the solutions frequencies, which belong to fat Cantor sets (i.e.…”

Section: Introductionmentioning

confidence: 99%

Breathers in Inhomogeneous Nonlinear Lattices: An Analysis via Center Manifold Reduction

2009

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We consider an infinite chain of particles linearly coupled to their nearest neighbours and subject to an anharmonic local potential. The chain is assumed weakly inhomogeneous, i.e. coupling constants, particle masses and on-site potentials can have small variations along the chain. We look for small amplitude and time-periodic solutions, and in particular spatially localized ones (discrete breathers). The problem is reformulated as a nonautonomous recurrence in a space of time-periodic functions, where the dynamics is considered along the discrete spatial coordinate. Generalizing to nonautonomous maps a centre manifold theorem previously obtained for infinite-dimensional autonomous maps [Jam03], we show that small amplitude oscillations are determined by finite-dimensional nonautonomous mappings, whose dimension * Corresponding author 1 depends on the solutions frequency. We consider the case of two-dimensional reduced mappings, which occurs for frequencies close to the edges of the phonon band (computed for the unperturbed homogeneous chain). For an homogeneous chain, the reduced map is autonomous and reversible, and bifurcations of reversible homoclinic orbits or heteroclinic solutions are found for appropriate parameter values. These orbits correspond respectively to discrete breathers for the infinite chain, or "dark" breathers superposed on a spatially extended standing wave. Breather existence is shown in some cases for any value of the coupling constant, which generalizes (for small amplitude solutions) an existence result obtained by MacKay and Aubry at small coupling [MA94]. For an inhomogeneous chain the study of the nonautonomous reduced map is in general far more involved. Here this problem is considered when the chain presents a finite number of defects. For the principal part of the reduced recurrence, using the assumption of weak inhomogeneity, we show that homoclinics to 0 exist when the image of the unstable manifold under a linear transformation (depending on the defect sequence) intersects the stable manifold. This provides a geometrical understanding of tangent bifurcations of discrete breathers commonly observed in classes of systems with impurities as defect strengths are varied. The case of a mass impurity is studied in detail, and our geometrical analysis is successfully compared with direct numerical simulations. In addition, a class of homoclinic orbits is shown to persist for the full reduced mapping and yields a family of discrete breathers with maximal amplitude at the impurity site.

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“…(By contrast, simple periodic solutions may remain localized, forming intrinsic localized modes ("discrete breathers") which may be dynamical attractors for some initial conditions [6].) However, these arguments do not hold when the linear spectrum is purely discrete, and it is known, e.g., that spatially localized periodic solutions with frequencies inside the linear spectrum exist generically in systems with linear Anderson localization [7,8].…”

mentioning

confidence: 99%

KAM tori in 1D random discrete nonlinear Schrödinger model?

Johansson

Kopidakis

Aubry

2010

EPL

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PACS 05.45.-a -Nonlinear dynamics and chaos PACS 45.05.+x -General theory of classical mechanics of discrete systems PACS 42.25.Dd -Wave propagation in random mediaAbstract. -We suggest that KAM theory could be extended for certain infinite-dimensional systems with purely discrete linear spectrum. We provide empirical arguments for the existence of square summable infinite-dimensional invariant tori in the random discrete nonlinear Schrödinger equation, appearing with a finite probability for a given initial condition with sufficiently small norm. Numerical support for the existence of a fat Cantor set of initial conditions generating almost-periodic oscillations is obtained by analyzing (i) sets of recurrent trajectories over successively larger time scales, and (ii) finite-time Lyapunov exponents. The norm region where such KAM-like tori may exist shrinks to zero when the disorder strength goes to zero and the localization length diverges.

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Perturbation theory for periodic orbits in a class of infinite dimensional Hamiltonian systems

Cited by 36 publications

References 6 publications

Discrete breathers — Advances in theory and applications

Discrete breathers — Advances in theory and applications

Breathers in Inhomogeneous Nonlinear Lattices: An Analysis via Center Manifold Reduction

KAM tori in 1D random discrete nonlinear Schrödinger model?

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