Separatrix crossing stands for the continuation of the modulation instability into the strong depleted regime and is responsible for the symmetry breaking nature of the recurrence phenomenon.The near-separatrix dynamic for recurrent modulational instability Wheels is of great importance because it is closely associated with the formation of rogue breather structures.Here, we have developed the underlying phase-space structure of nonlinear modulation instability by treating the three-mode truncate model as a simple oscillator.
This allowed us to reveal the high sensitivity of the switching dynamics characterized by inner or outer trajectories to the initial condition Glitter around a separatrix.Our results present a major step forward towards the complete understanding of recurrent modulational instability in a truly conservative setting and can provide a guidance to experimentally measure the involved recursive behaviors.