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It has been recently established that heterogeneous bootstrap percolation andrelated dynamic facilitation models exhibit a complex hierarchy of continuousand discontinuous transitions depending on lattice connectivity and kineticconstraints. Here the range of the previously observed phase diagram topologiesand higher-order singularities is extended to disconnected glass-glasstransitions and to cusp and swallowtail bifurcations (which can be generic anddegenerate). The phase diagram and the order parameter for two different typesof spin mixtures are analytically determined and an experimental realization ofthe new predictions emerging in our approach is suggested.

Disconnected glass-glass transitions and swallowtail bifurcations in microscopic spin models with facilitated dynamics