This letter investigates the period-doubling cascades of canards, generated in the extended Bonhoeffer-van der Pol oscillator. Canards appear by Andronov Hopf bifurcations (AHBs) and it is confirmed that these AHBs are always supercritical in our system. The cascades of period-doubling bifurcation are followed by mixed-mode oscillations. The detailed two-parameter bifurcation diagrams are derived, and it is clarified that the period-doubling bifurcations arise from a narrow parameter value range at which the original canard in the non-extended equation is observed.

No abstract

In this paper, we analyze circuits which exhibit a torus doubling route to chaos. Though some researchers insisted that torus doubling occurs a finite number of times, we show that the successive occurrence of a certain kind of torus doubling, which we call "swollen shape type bifurcation," is observed in a certain region. This phenomenon has a very delicate structure, and when observed in a laboratory experiment is very interesting. In addition, a coupled system of a logistic map and a sin circle map is proposed as the simplest model which can explain the mechanism of the swollen shape type bifurcation. Also, the swollen shape type bifurcation is also observed in this system.

No abstract

No abstract