Collapse of mixed-mode oscillations and chaos in the extended Bonhoeffer-van Der pol oscillator under weak periodic perturbation

Mixed-mode oscillations in a slow-fast dynamical system under weak perturbation are studied numerically. First, we make a band-limited extremely weak Gaussian noise, and apply this noise to this oscillator. Then, we observe random phenomenon from numerical study even if the noise is extremely weak. The mixed-mode oscillations are submerged by chaos due to extremely weak noise. We imagine that mixed-mode oscillations in a slow-fast systems are delicate to the noise. In order to make clear the mechanism of generation of chaos, we assume that weak perturbation is periodic. From this assumption, we can calculate Lyapunov exponent, and draw a bifurcation diagram. In this bifurcation diagram, period-doubling bifurcations take place when the amplitude of the periodic perturbation is extremely small. We suspect the observability of the mixed-mode oscillation of the slow-fast dynamical system by experiment from this numerical result.