Bushes of vibrational modes for Fermi–Pasta–Ulam chains

Chechin, G. M.; Novikova, Natalya; Abramenko, A. A.

doi:10.1016/s0167-2789(02)00430-x

Cited by 83 publications

(140 citation statements)

References 20 publications

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“…A more systematic method for finding invariant manifolds in a physical system should be based on the symmetries of this system. The only reference that exploits these symmetries for the FPU lattice is [5] in which so-called 'bushes of normal modes' are computed. These 'bushes' are simply invariant manifolds of a certain type.…”

Section: In Whichmentioning

confidence: 99%

Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice

Rink

2003

Physica D: Nonlinear Phenomena

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The Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and n particles admits a large group of discrete symmetries. The fixed point sets of these symmetries naturally form invariant symplectic manifolds that are investigated in this short note. For each k dividing n we find k degree of freedom invariant manifolds. They represent short wavelength solutions composed of k Fourier-modes and can be interpreted as embedded lattices with periodic boundary conditions and only k particles. Inside these invariant manifolds other invariant structures and exact solutions are found which represent for instance periodic and quasi-periodic solutions and standing and traveling waves. Some of these results have been found previously by other authors via a study of mode coupling coefficients and recently also by investigating 'bushes of normal modes'. The method of this paper is similar to the latter method and much more systematic than the former. We arrive at previously unknown results without any difficult computations. It is shown moreover that similar invariant manifolds exist also in the Klein-Gordon lattice and in the thermodynamic and continuum limits.

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Section: In Whichmentioning

confidence: 99%

Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice

Rink

2003

Physica D: Nonlinear Phenomena

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“…These results were obtained using a direct linear stability analysis around the periodic orbit corresponding to the π-mode. Similar methods have been more recently applied to other modes and other FPU potentials by Chechin [71,72] and Rink [73]. A technique which allows for a more general exploration of the dynamics starting from short wavelengths is to follow an envelope function of the oscillators defined by ψ i = (−1) i q i .…”

Section: Dynamics At Short Wavelengths: Chaotic Breathersmentioning

confidence: 99%

“…Such a set is an invariant manifold in Fourier space. The question of existence has been positively solved [70,71]. For instance, modes k = N 4 ; N 3 ; N 2 ; 2N 3 ; 3N 4 (2.73)…”

Section: Short-wavelength (High-frequency) Initial Conditionsmentioning

confidence: 99%

“…Moreover periodic and quasi-periodic solutions evolving on two-mode manifolds have been derived, and a full classification of higher dimensional invariant manifolds has been obtained. The existence of these invariant manifolds is related to spatial symmetries [71,72]. The question of linear stability is more difficult and it has been solved analytically only for the zone boundary mode k = N/2.…”

Section: Short-wavelength (High-frequency) Initial Conditionsmentioning

confidence: 99%

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Dynamics of Oscillator Chains

Lichtenberg

Livi

Pettini

et al.

The Fermi-Pasta-Ulam Problem

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Abstract. The Fermi-Pasta-Ulam (FPU) nonlinear oscillator chain has proved to be a seminal system for investigating problems in nonlinear dynamics. First proposed as a nonlinear system to elucidate the foundations of statistical mechanics, the initial lack of confirmation of the researchers expectations eventually led to a number of profound insights into the behavior of high-dimensional nonlinear systems. The initial numerical studies, proposed to demonstrate that energy placed in a single mode of the linearized chain would approach equipartition through nonlinear interactions, surprisingly showed recurrences. Although subsequent work showed that the origin of the recurrences is nonlinear resonance, the question of lack of equipartition remained. The attempt to understand the regularity bore fruit in a profound development in nonlinear dynamics: the birth of soliton theory. A parallel development, related to numerical observations that, at higher energies, equipartition among modes could be approached, was the understanding that the transition with increasing energy is due to resonance overlap. Further numerical investigations showed that time-scales were also important, with a transition between faster and slower evolution. This was explained in terms of mode overlap at higher energy and resonance overlap at lower energy. Numerical limitations to observing a very slow approach to equipartition and the problem of connecting high-dimensional Hamiltonian systems to lower dimensional studies of Arnold diffusion, which indicate transitions from exponentially slow diffusion along resonances to power-law diffusion across resonances, have been considered. Most of the work, both numerical and theoretical, started from low frequency (long wavelength) initial conditions. Coincident with developments to understand equipartition was another program to connect a statistical phenomenon to nonlinear dynamics, that of understanding classical heat conduction. The numerical studies were quite different, involving the excitation of a boundary oscillator with chaotic motion, rather than the excitation of

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“…Other authors, cf. [5], have baptized these invariant manifolds bushes of normal modes. The restriction of a Hamiltonian vector field to a fixed point set of a group is often easy to compute:…”

Section: Discrete Symmetrymentioning

confidence: 99%

An Integrable Approximation for the Fermi–Pasta–Ulam Lattice

Rink

The Fermi-Pasta-Ulam Problem

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This contribution presents a review of results obtained from computations of approximate equations of motion for the Fermi-Pasta-Ulam lattice. These approximate equations are obtained as a finite-dimensional Birkhoff normal form. It turns out that in many cases, the Birkhoff normal form is suitable for application of the KAM theorem. In particular this proves Nishida's 1971 conjecture stating that almost all low-energetic motions of the anharmonic Fermi-Pasta-Ulam lattice with fixed endpoints are quasi-periodic. The proof is based on the formal Birkhoff normal form computations of Nishida, the KAM theorem and discrete symmetry considerations.

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Bushes of vibrational modes for Fermi–Pasta–Ulam chains

Cited by 83 publications

References 20 publications

Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice

Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice

Dynamics of Oscillator Chains

An Integrable Approximation for the Fermi–Pasta–Ulam Lattice

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