Energy localization on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>q</mml:mi></mml:math>-tori, long-term stability, and the interpretation of Fermi-Pasta-Ulam recurrences

Christodoulidi, Helen; Efthymiopoulos, Christos; Bountis, Tassos

doi:10.1103/physreve.81.016210

Cited by 60 publications

(91 citation statements)

References 38 publications

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“…corresponding to the full quasi-periodic case is beyond the scope of the present paper and is left to future investigation (see however Refs. [44,45] for interesting results in this direction). In order to describe the approach to equipartition in the FPU system, in Fig.…”

Section: Toda Equilibrium and Quasi-statementioning

confidence: 95%

The two-stage dynamics in the Fermi-Pasta-Ulam problem: From regular to diffusive behavior

Ponno

Christodoulidi

Skokos

et al. 2011

Chaos: An Interdisciplinary Journal of Nonlinear Science

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A numerical and analytical study of the relaxation to equilibrium of both the FermiPasta-Ulam (FPU) α-model and the integrable Toda model, when the fundamental mode is initially excited, is reported. We show that the dynamics of both systems is almost identical on the short term, when the energies of the initially unexcited modes grow in geometric progression with time, through a secular avalanche process. At the end of this first stage of the dynamics the time-averaged modal energy spectrum of the Toda system stabilizes to its final profile, well described, at low energy, by the spectrum of a q-breather. The Toda equilibrium state is clearly shown to describe well the long-living quasi-state of the FPU system. On the long term, the modal energy spectrum of the FPU system slowly detaches from the Toda one by a diffusive-like rising of the tail modes, and eventually reaches the equilibrium flat shape. We find a simple law describing the growth of tail modes, which enables us to estimate the time-scale to equipartition of the FPU system, even when, at small energies, it becomes unobservable.

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Section: Toda Equilibrium and Quasi-statementioning

confidence: 95%

The two-stage dynamics in the Fermi-Pasta-Ulam problem: From regular to diffusive behavior

Ponno

Christodoulidi

Skokos

et al. 2011

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

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“…Similarly a 2 and a 4 are invariant under [1,3] and exchanged by [2,4]. The total solution (a 1 , a 2 , a 3 , a 4 ) is changed by non-trivial elements of D 4 , except in cases with additional symmetry.…”

Section: A22 Computation Of Fibersmentioning

confidence: 99%

“…In the last stage we determine a 1 and a 3 , invariant under [2,4] and exchanged by [1,3]. Similarly a 2 and a 4 are invariant under [1,3] and exchanged by [2,4].…”

Section: A22 Computation Of Fibersmentioning

confidence: 99%

“…(If a point a = (a 1 , a 2 , a 3 , a 4 ) of the fiber can be found then η 1 = q 1 (a) and η 2 = q 2 (a), hence invariant under D 4 .) The quantity 4(−a 1 +a 2 −a 3 +a 4 ) = √ 1 − 16η 2 is invariant under the subgroup V 4 ⊂ D 4 generated by the permutations [1,3] and [2,4], and sent to its negative by [1,2] [3,4]. The group V 4 also leaves invariant the quantities s 13 = a 1 + a 3 , s 24 = a 2 + a 4 , p 13 = a 1 a 3 , and p 24 = a 2 a 4 .…”

Section: A22 Computation Of Fibersmentioning

confidence: 99%

“…2. There exists a huge amount of literature on the FPU chain but nearly always regarding the classical case of equal masses, see for recent references [4]. One should note that, although the classical case looks physically natural, its dynamics is non-generic because of symmetries.…”

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confidence: 99%

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The Inhomogeneous Fermi-Pasta-Ulam Chain, a Case Study of the 1 : 2 : 3 $1:2:3$ Resonance

Bruggeman

Verhulst

2017

Acta Appl Math

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A 4-particles chain with different masses represents a natural generalization of the classical Fermi-Pasta-Ulam chain. It is studied by identifying the mass ratios that produce prominent resonances. This is a technically complicated problem as we have to solve an inverse problem for the spectrum of the corresponding linearized equations of motion. In the case of such an inhomogeneous periodic chain with four particles each mass ratio determines a frequency ratio for the quadratic part of the Hamiltonian. Most prominent frequency ratios occur but not all. In general we find a one-dimensional variety of mass ratios for a given frequency ratio.A detailed study is presented of the resonance 1 : 2 : 3. A small cubic term added to the Hamiltonian leads to a dynamical behaviour that shows a difference between the case that two opposite masses are equal and a striking difference with the classical case of four equal masses. For two equal masses and two different ones the normalized system is integrable and chaotic behaviour is small-scale. In the transition to four different masses we find a Hamiltonian-Hopf bifurcation of one of the normal modes leading to complex instability and Shilnikov-Devaney bifurcation. The other families of short-periodic solutions can be localized from the normal forms together with their stability characteristics. For illustration we use action simplices and examples of behaviour with time.

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Variations on the Fermi-Pasta-Ulam Chain, a Survey

Verhulst

2021

13th Chaotic Modeling and Simulation International Conference

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Energy localization onq-tori, long-term stability, and the interpretation of Fermi-Pasta-Ulam recurrences

Cited by 60 publications

References 38 publications

The two-stage dynamics in the Fermi-Pasta-Ulam problem: From regular to diffusive behavior

The two-stage dynamics in the Fermi-Pasta-Ulam problem: From regular to diffusive behavior

The Inhomogeneous Fermi-Pasta-Ulam Chain, a Case Study of the 1 : 2 : 3 $1:2:3$ Resonance

Variations on the Fermi-Pasta-Ulam Chain, a Survey

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