Unveiling Complex Oscillations and Bursting Synchronization in the Extended Bonhoeffer–van der Pol System

ABSTRACT This study centers on the extended Bonhoeffer–van der Pol (EBVP) electronic circuit, in which the complex oscillations exhibit intricate multi‐timescale characteristics. By means of slow‐fast decomposition techniques, we conduct a comprehensive exploration of the oscillation behaviors and bifurcation mechanisms. Specifically, we discover that the fast dynamics trigger abrupt transitions modulated by the system's slow components. This coupling gives rise to diverse phenomena, including significant amplitude variations and the emergence of complex waveforms. We construct a singular periodic orbit by leveraging the time‐scale ratio and further analyze the period‐adding sequences in the EBVP system with one slow and two fast variables. Subsequently, we implement adaptive synchronization in the system based on the backstepping design. Numerical simulations are carried out to validate the existence of bursting oscillations and the effectiveness of the synchronization process.