The bifurcation structure of fractional-harmonic entrainments in the forced Rayleigh oscillator

The bifurcation structure of fractional harmonic entrainments in the forced Rayleigh oscillator is studied. In each fractional harmonic entrainment, complex bifurcation occurs when the saddle node bifurcation curve, which forms fractional harmonic entrainments, approaches the Neimark-Sacker bifurcation curve which follows the fundamental harmonic entrainment. Two types of fractional harmonic entrainments are observed. We call the entrainments Type I and Type II. In Type I, the attractor which is asymmetric with respect to the origin is borne by a saddle node bifurcation. Subsequently, period doubling bifurcation occurs and chaos is generated. In Type II, the attractor which is symmetric with respect to the origin is borne by a saddle node bifurcation. Symmetry breaking transition occurs in the tube of the saddle node bifurcation curve. Subsequently, period doubling bifurcation occurs and chaos is generated