Modeling the collapse of the edge when two transition routes compete

The transition to turbulence in many shear flows proceeds along two competing routes, one linked with finite-amplitude disturbances and the other one originating from a linear instability, as in, e.g., boundary layer flows. The dynamical systems concept of an edge manifold has been suggested in the subcritical case to explain the partition of the state space of the system. This investigation is devoted to the evolution of the edge manifold when linear stability is added in such subcritical systems, a situation poorly studied despite its prevalence in realistic fluid flows. In particular, the fate of the edge state as a mediator of transition is unclear. A deterministic three-dimensional model is suggested, parametrized by the linear instability growth rate. The edge manifold evolves topologically, via a global saddle-loop bifurcation of the underlying invariant sets, from the separatrix between two attraction basins to the mediator between two transition routes. For larger instability rates, the stable manifold of the saddle point increases in codimension from 1 to 2 after an additional local pitchfork node bifurcation, causing the collapse of the edge manifold. As the growth rate is increased, three different regimes of this model are identified, each one associated with a flow case from the recent hydrodynamic literature. A simple nonautonomous generalization of the model is also suggested in order to capture the complexity of spatially developing flows.