Hopf bifurcation with \(D_ 3\times D_ 3\)-symmetry
The authors investigate a generic Hopf bifurcation of a hierarchical structure of three interacting sets of three oscillators. The system is assumed to be symmetric with respect to permutation of the three groups and of the individual oscillators. The Hopf bifurcation is investigated by computing isotropy subgroups and fixed point subspaces of the eight dimensional center eigenspace. The normal form admits 11 distinct periodic solutions with maximal isotropy subgroups; these are classified and their branching-types and stabilities are determined in terms of the cubic and relevant quintic coefficients of the normal form. The possibility of quasiperiodic solutions and heteroclinic cycles is discussed. The paper demonstrates that by simplifying the model to the ``macro- level'' the remaining symmetries from the ``micro-level'' may lead to hidden symmetries, which cannot be interpreted by observing only the macro-variables.
three interacting sets of three oscillators
scientific article; zbMATH DE number 431194