Dynamics of a Chain with Four Particles, Alternating Masses and Nearest-Neighbor Interaction

Bruggeman, Roelof W.; Verhulst, Ferdinand

doi:10.1007/978-3-319-63937-6_5

Cited by 3 publications

(5 citation statements)

References 9 publications

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“…These solutions are unstable for the α-chain, and stable for the β-chain, see again Bruggeman and Verhulst (2017b). At this level of approximation we find weak interaction between the optical and the acoustical group, and no equipartition.…”

Section: Four Alternating Masses a Summarymentioning

confidence: 63%

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Near-Integrability and Recurrence in FPU Chains with Alternating Masses

Bruggeman

Verhulst

2018

J Nonlinear Sci

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An important aspect of understanding FPU chains is the existence of invariant manifolds (called "bushes") in FPU chains. We will focus on the classical periodic FPU chain and on the FPU chain with alternating masses where we show that also in the alternating case nested manifolds (related to bushes) exist. The use of symmetries leads to the emergence of systems of n particles as invariant manifolds of systems with a multiple of n particles. This analysis is followed by examples of existence and stability of special invariant manifolds and phase-space dynamics in the case of 4 and 8 particles. These examples are typical for periodic FPU chains with 4n or 8n particles. It turns out that in the alternating case the dynamics is strongly affected by the choice of the alternating mass m. Normal form calculations help to identify quasi-trapping regions leading to delay of recurrence. The results suggest that equipartition of energy near stable equilibrium is improbable.

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Section: Four Alternating Masses a Summarymentioning

confidence: 63%

“…This case has been analyzed in Bruggeman and Verhulst (2017b). The dynamics of this case will be found again in systems with 8 particles, in general 4n particles.…”

Section: Four Alternating Masses a Summarymentioning

confidence: 99%

Near-Integrability and Recurrence in FPU Chains with Alternating Masses

Bruggeman

Verhulst

2018

J Nonlinear Sci

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“…( 10) suggest that for mass m large we have two groups of oscillators, one with frequency size close to √ 2 and one with size O( √ a). There are indications in Bruggeman and Verhulst [3] that in the case of a chain with 4 particles there exists significant interactions between the 2 groups. It turns out that in α-chains the acoustical group can be strongly excited by the optical group.…”

Section: [3]mentioning

confidence: 99%

“…The modes x 1 and x 2 are in a detuned 1 : 1 resonance when choosing 0 < a 1 . Consider the general position periodic solution of the 1 : 1 resonance of the x 1 , x 2 modes, described in Bruggeman and Verhulst [3]. A normal form approximation is x 1 (t) = r 0 cos( √ 2t + ψ 0 ), x 1 (t) = ±x 2 (t); the approximation is based on the equations for these modes to order O(a):…”

Section: Interactions Between Optical and Acoustical Groupmentioning

confidence: 99%

Variations on the Fermi-Pasta-Ulam chain, a survey

Verhulst¹

2020

Preprint

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We will present a survey of low energy periodic Fermi-Pasta-Ulam chains with leading idea the "breaking of symmetry". The classical periodic FPU-chain (equal masses for all particles) was analysed by Rink in 2001 with main conclusions that the normal form of the beta-chain is always integrable and that in many cases this also holds for the alfa-chain. The implication is that the KAM-theorem applies to the classical chain so that at low energy most orbits are located on invariant tori and display quasi-periodic behavior. Most of the reasoning also applies to the FPU-chain with fixed endpoints.The FPU-chain with alternating masses already shows a certain breaking of symmetry. Three exact families of periodic solutions can be identified and a few exact invariant manifolds which are related to the results of Chechin et al. (1998Chechin et al. ( -2005 on bushes of periodic solutions. An alternating chain of 2n particles is present as submanifold in chains with k 2n particles, k=2, 3, ... The normal forms are strongly dependent on the alternating masses 1, m, 1, m,... If m is not equal to 2 or 4/3 the cubic normal form of the Hamiltonian vanishes. For alfa-chains there are some open questions regarding the integrability of the normal forms if m= 2 or 4/3. Interaction between the optical and acoustical group in the case of large mass m is demonstrated.The part played by resonance suggests the role of the mass ratios. It turns out that in the case of 4 particles there are 3 first order resonances and 10 second order ones; the 1:1:1:...:1 resonance does not arise for any number of particles and mass ratios. An interesting case is the 1:2:3 resonance that produces after a Hamilton-Hopf bifurcation and breaking symmetry chaotic behaviour in the sense of Shilnikov-Devaney. Another interesting case is the 1:2:4 resonance. As expected the analysis of various cases has a significant impact on recurrence phenomena; this will be illustrated by numerical results.

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“…The invariant tori obstruct or at least delay equipartition of energy, see also [19]. It turns out that such integrable approximations do also exist for the fpu-chain with alternating masses [11,5], see in particular [4] where the case m j = 1, m, 1, m of four alternating masses is analysed.…”

Section: Introductionmentioning

confidence: 96%

The 1:2:4 resonance in a particle chain

Hanßmann¹,

Mazrooei‐Sebdani²,

Verhulst³

2020

Preprint

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We consider four masses in a circular configuration with nearest-neighbour interaction, generalizing the spatially periodic Fermi-Pasta-Ulam-chain where all masses are equal. We identify the mass ratios that produce the 1:2:4 resonance -the normal form in general is non-integrable already at cubic order. Taking two of the four masses equal allows to retain a discrete symmetry of the fully symmetric Fermi-Pasta-Ulam-chain and yields an integrable normal form approximation. The latter is also true if the cubic terms of the potential vanish. We put these cases in context and analyse the resulting dynamics, including a detuning of the 1:2:4 resonance within the particle chain.

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Dynamics of a Chain with Four Particles, Alternating Masses and Nearest-Neighbor Interaction

Cited by 3 publications

References 9 publications

Near-Integrability and Recurrence in FPU Chains with Alternating Masses

Near-Integrability and Recurrence in FPU Chains with Alternating Masses

Variations on the Fermi-Pasta-Ulam chain, a survey

The 1:2:4 resonance in a particle chain

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