Analysis of mixed-mode oscillation-incrementing bifurcations generated in a nonautonomous constrained Bonhoeffer-van der Pol oscillator

Mixed-mode oscillations (MMOs) are phenomena observed in a number of dynamic settings, including electrical circuits and chemical systems. Mixed-mode oscillation-incrementing bifurcations (MMOIBs) are among the most complex MMO bifurcations observed in the large group of MMO-generating dynamics; however, only a few theoretical analyses of the mechanism causing MMOIBs have been performed to date. In this study, we use a degenerate technique to analyze MMOIBs generated in a Bonhoeffer-van der Pol oscillator with a diode under weak periodic perturbation. We consider the idealized case in which the diode operates as an ideal switch; in this case, the governing equation of the oscillator is a piecewise smooth constraint equation and the Poincaré return map is one-dimensional, and we find that MMOIBs occur in a manner similar to period-adding bifurcations generated by the circle map. Our numerical results suggest that the universal constant converges to 1.0 and our experimental results demonstrate that MMOIBs can occur successively many times. Our one-dimensional Poincaré return map clearly answers the fundamental question of why MMOs are related to Farey sequences even though each MMO-generating region in the parameter space is terminated by chaos.