Nonautonomous scalar concave-convex differential equations: Conditions for uniform stability or bistability in a...

We investigate the long-term dynamics of a nonautonomous Bonifacio-Lugiato model of optical superfluorescence. The scalar ordinary differential equation modeling the phenomenon is given by a concave-convex autonomous function of the state variable that is excited by a time-dependent input, Λ(t). We describe the system's response in terms of the dynamical characteristics of the input function, with particular focus on the cases of uniform stability-when exactly a bounded solution exists, which in addition is hyperbolic attractive-or bistability-when two stable solutions of this type coexist. Our starting point is the open interval delimited by the constant input values λ for which the autonomous version of our model was already known to exhibit bistability: we prove that, in general, bistability occurs when Λ(t) lies within this interval. This condition is sufficient but not necessary. Applying nonautonomous bifurcation methods and imposing more restrictive conditions on the variation of Λ(t), we can determine the necessary and sufficient conditions for bistability and to prove that the general response is uniform stability when these conditions are not satisfied. Finally, we analyze the case of a periodic input that varies on a slow timescale using fast-slow system methods to rigorously establish either a uniformly stable or a bistable response.