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Paper   IPM / M / 17559
School of Mathematics
  Title:   Hamiltonian Hopf bifurcations near a chaotic Hamiltonian resonance
  Author(s):  Reza Mazrooei-Sebdani (Joint with E. Hakimi)
  Status:   Published
  Journal: Physica D: Nonlinear Phenomena
  Vol.:  459
  Year:  2024
  Pages:   134017
  Supported by:  IPM
The 1:2:3 Hamiltonian resonance is one of the four genuine first order resonances which is non-integrable. For this resonance, chaotic behavior of the normal form has been shown due to the existence of a transverse homoclinic orbit on the energy manifold. Considering the detuning parameters, in the mirror symmetric cases of the Poisson manifold, by a reduction theory and computing the classical normal form of non-degenerate Hamiltonian Hopf bifurcation, we show there are just non-degenerate Hamiltonian Hopf bifurcations. However in the general case by considering some case studies and specially of FPU (Fermi–Pasta–Ulam) chains for a fiber passaging 1:2:3 resonance, we may deal with complex dynamics and specially Hamiltonian Hopf bifurcation in a complicated region. Actually, we can see some special Krein collisions in a complex region.

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