Explosive entrainment in Arnold tongues

Entrainment refers to the unidirectional synchronization of an oscillator system to external periodic forcing, with phase-locking regions forming Arnold tongues. In some specific models, the hysteresis phenomenon of Arnold tongue has been observed, but methods to actively induce it across models are still lacking. In this study, we show that adaptive responses to periodic forcing can induce explosive entrainment, a phenomenon of sudden locking to an external frequency accompanied by significant hysteresis. In contrast to explosive synchronization, where synchronization emerges suddenly due to internal coupling, explosive entrainment describes the sudden locking to a given frequency caused by external forcing. Therefore, explosive entrainment is more likely to occur in systems where internal couplings are weak or difficult to estimate, but external forcing can be applied. Furthermore, we demonstrate the universal emergence of explosive entrainment in the Kuramoto model, Poincaré model, chaotic Rössler model, and complex Ginzburg-Landau equation. Theoretical hysteresis bounds in the Kuramoto model are determined based on analytical results of the Ott-Antonsen reduction method. This study provides ideas for actively inducing hysteresis through external forcing.