Mechanism for suppression of chaos by weak resonant perturbations: part II

Suppression of chaos by weak resonant perturbations has been a hot topic in recent years. Nobody has answered the most important problem why chaos disappears for weak resonant perturbations. For this problem, the authors showed theoretically in a previous paper that one of the possible roots to suppression of chaos is a saddle node bifurcation [Phys. Rev. Lett., vol. 83, p. 3824, (1999)]. In this paper, the mechanism of another root to suppression of chaos is clarified theoretically by the use of degeneration technique. The model studied in this paper is a forced Rayleigh oscillator with a diode. By weak resonant perturbations, a chaotic attractor with a wide band becomes a chaotic attractor with a narrow band, and a periodic attractor. We consider the case where the diode in the circuit operates as a switch. In this case, the Poincare map is derived rigorously as a one-dimensional mapping. By the analysis of the map, it is clarified theoretically that the transition from chaos with a wide band to chaos with a narrow band is crisis and that the transition from chaos with a narrow band to a periodic attractor is an inverse period doubling bifurcation