Emergence of Extreme Events in Weakly Coupled Autonomous Bonhoeffer–van der Pol Systems

In this paper, we report, for the first time, that Extreme Events (EEs) can robustly emerge in autonomous, weakly coupled Bonhoeffer–van der Pol oscillators, challenging the prevailing notion that strong interaction and external drive are necessary for such rare, large-amplitude bursts. Our central finding is a novel dynamical route to EEs: a transition from quasi-periodic torus dynamics to EEs via torus breakdown induced by interior crisis, triggered purely by infinitesimal diffusive coupling and local instability. Stability analysis reveals how subtle nonlinear slope variations in the weak-coupling regime destabilize the torus, whereas stronger coupling suppresses EEs and restores classical synchronization. Using a suite of diagnostics such as time series, phase portraits, bifurcation diagrams, Poincaré sections, and statistical measures like probability density functions and inter-event intervals, we rigorously confirm the intermittent, heavy-tailed nature of the events. A two-parameter scan identifies precise regions of EE prevalence, and mechanistic analysis links the onset to eigenvalue topology changes and crisis-induced attractor expansion. This work establishes torus breakdown under weak coupling as a fundamental, noise-free pathway to EEs, with broad implications for predicting rare dynamics in physical, biological, and engineered networked systems.