TY - JOUR

T1 - The 1:2:4 resonance in a particle chain

AU - Hanßmann, H.

AU - Mazrooei-Sebdani, R.

AU - Verhulst, F.

N1 - Publisher Copyright:
© 2020 The Authors

PY - 2021/2/1

Y1 - 2021/2/1

N2 - We consider four masses in a circular configuration with nearest-neighbour interaction, generalising 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.

AB - We consider four masses in a circular configuration with nearest-neighbour interaction, generalising 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.

KW - Fermi–Pasta–Ulam chain

KW - Integrability

KW - Normal forms

KW - Resonance

UR - http://www.scopus.com/inward/record.url?scp=85086372314&partnerID=8YFLogxK

U2 - 10.1016/j.indag.2020.06.003

DO - 10.1016/j.indag.2020.06.003

M3 - Article

AN - SCOPUS:85086372314

SN - 0019-3577

VL - 32

SP - 101

EP - 120

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

IS - 1

ER -