Implications of tristability on localization phenomena: A necking bifurcation’s tale

We analyze the implication of tristability on localization phenomena in one-dimensional extended dissipative systems. In this context, localized states appear due to the interaction and locking of front waves connecting different extended states. In the tristable regime investigated here two extended uniform states coexist with one periodic Turing pattern. This scenario leads to the transition from the standard-homoclinic-snaking-related localized states associated with uniform-pattern bistability to the collapsed-homoclinic-snaking-related states which arise in a uniform-bistable configuration. We find that this transition is mediated by the emergence of hybrid states through codimension-two necking bifurcations. To perform this study we use bifurcation analysis on a non-variational mean-field model describing the spatiotemporal dynamics of light pulses in passive Kerr cavities. •Our work investigates the formation of localized dissipative structures in extended tristable systems.•We report a complex, but smooth, transition between standard- homoclinic-snaking- and collapsed-homoclinic-snaking-related localized states.•This transition is mediated by a cascade of necking bifurcations which organize the parameter space of these types of systems.•Our results confirm that this transition is universal in uniform-pattern-uniform tristable systems.