Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system

Cafarella, Alessandro; Leo, M.; Leo, R. A.

doi:10.1103/physreve.69.046604

Cited by 31 publications

(26 citation statements)

References 24 publications

(47 reference statements)

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“…The asymptotic behavior for large N of this formula gives the same threshold as (2.99) in Sect. 2.4.6, see also [81,139]. This critical energy is also very close to the Chirikov threshold for short wavelength (2.7).…”

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

confidence: 73%

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

confidence: 73%

Dynamics of Oscillator Chains

Lichtenberg

Livi

Pettini

et al.

The Fermi-Pasta-Ulam Problem

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“…This work is similar to ours in that the eigenvectors of the linearized problem are used to change variables and simplify the original N-dimensional problem. Continuing in this vein, we find [20] uses symplectic numerical methods to study the nonlinear stability of all NNMs found by [19]; [21] studies ZBM stability in the case β < 0 using the method of averaging; and [22] studies NNM stability with wavelength equal to four times the lattice spacing, the so-called ''π /2 mode''. Note that these authors refer to NNMs as one-mode solutions (OMSs).…”

Section: A2 Related Work On Fpu Systemsmentioning

confidence: 99%

The zone boundary mode in periodic nonlinear electrical lattices

Bhat

Osting

2009

Physica D: Nonlinear Phenomena

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“…Let us note that stability of the π-mode in the FPU-α and FPU-β chains was investigated by different methods in a large number of papers (see, for example, [18,19,20,8,9,10,12,13,14,15]), but, to our best understanding, the influence of symmetry of these mechanical models on stability analysis was not discussed. Unlike the above cited works, the bush stability analysis presented in this paper based on the symmetry-related arguments only.…”

Section: Consideringmentioning

confidence: 99%

“…Note, that the stability of the π-mode (the bush B[â 2 ,î]) was discussed in a number of papers [18,19,20,8,9,10,12,13,14,15] by different methods and with an emphasis on different aspects of this stability. In particular, in our paper [8], a remarkable fact was revealed for the FPU-α chain: the stability threshold of the π-mode is one and the same for interactions with all the other modes of the chain.…”

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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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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Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system

Cited by 31 publications

References 24 publications

Dynamics of Oscillator Chains

Dynamics of Oscillator Chains

The zone boundary mode in periodic nonlinear electrical lattices

Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries

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