Anomalous Hopf Bifurcations in Symmetrically-coupled Period-doubling Systems

초록

We consider two symmetrically-coupled logistic maps, and investigate the effect of symmetry on Hopf bifurcations, giving rise to the birth of the daughter orbits encircling the symmetric anti-phase mother orbit (with a time shift of half a period). When the rotation numbers v of daughter orbits are rational (i.e., v = r/s; r and s: coprimes), anomalous Hopf bifurcations are found to occur due to the symmetry of the coupled system. For even r, a symmetric periodic attractor is born through a standard Hopf bifurcation while for odd r, a conjugate pair of asymmetric periodic attractors appears via a nonstandard double Hopf bifurcation. These symmetry-conserving and -breaking Hopf bifurcations are explained by using the concept of a half-cycle map of a symmetrically-coupled map. These anomalous Hopf bifurcations might be observed in real experiments of symmetrically-coupled period-doubling systems.

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