Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice

Rink, Bob

doi:10.1016/s0167-2789(02)00694-2

Cited by 57 publications

(70 citation statements)

References 13 publications

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“…The invariant manifolds (bushes of modes) in the FPU-β model with respect to the group G ′ 0 were found by Rink in [11]. Some additional details of this problem were discussed in our paper [9] (in particular, the dynamical equations and stability of these bushes of modes).…”

Section: B Stability Of the π-Mode In The Fpu-β Chainmentioning

confidence: 67%

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Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries

Chechin

Zhukov

2006

Phys. Rev. E

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We present a theorem that allows to simplify linear stability analysis of periodic and quasiperiodic nonlinear regimes in N -particle mechanical systems (both conservative and dissipative) with different kinds of discrete symmetry. This theorem suggests a decomposition of the linearized system arising in the standard stability analysis into a number of subsystems whose dimensions can be considerably less than that of the full system. As an example of such simplification, we discuss the stability of bushes of modes (invariant manifolds) for the Fermi-Pasta-Ulam chains and prove another theorem about the maximal dimension of the above mentioned subsystems.

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Section: B Stability Of the π-Mode In The Fpu-β Chainmentioning

confidence: 67%

“…It is easy to check that both transformations (11) and (12) produce systems of equations which are equivalent to the system (9). Moreover, these transformations act on the individual equations u j (j = 1, 2, 3, 4) of the system (9) exactly as on the components x j (j = 1, 2, 3, 4)…”

Section: X(t) = {0 A(t) B(t) 0 −B(t) −A(t) | }mentioning

confidence: 99%

Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries

Chechin

Zhukov

2006

Phys. Rev. E

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“…Finally, we do not address here the symmetric invariant manifolds in the KG lattices because they can be retrieved from [14].…”

Section: Discussionmentioning

confidence: 99%

Near-Integrability of Low-Dimensional Periodic Klein-Gordon Lattices

Christov

2018

Advances in Mathematical Physics

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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%

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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Symmetric invariant manifolds in the Fermi–Pasta–Ulam lattice

Cited by 57 publications

References 13 publications

Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries

Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries

Near-Integrability of Low-Dimensional Periodic Klein-Gordon Lattices

Dynamics of Oscillator Chains

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