Nonautonomous scalar concave-convex differential equations: conditions for uniform stability or bistability in a model of optical fluorescence

TL;DR

This paper analyzes the long-term dynamics of a scalar differential equation modeling optical superfluorescence, establishing conditions for uniform stability and bistability based on input variations, including periodic inputs.

Contribution

It provides necessary and sufficient conditions for bistability and stability in a nonautonomous scalar differential equation with concave-convex nonlinearity, extending previous results.

Findings

Bistability occurs when input lies within a specific interval.

Conditions for uniform stability are established for certain input variations.

Periodic slow-varying inputs lead to either stable or bistable responses.

Abstract

The long-term dynamics of a Bonifacio-Lugiato model of optical superfluorescence is investigated. The scalar ordinary differential equation modelling the phenomenon is given by a concave-convex autonomous function of the state variable that is excited by a time-dependent input, $I(t)$. The system's response is described in terms of the dynamical characteristics of the input function, with particular focus on uniform stability or bistability cases. Building on previous published results, the open interval defined by the constant input values for which the equation exhibits uniform stability or bistability is considered, and it is proved that bistability occurs when $I(t)$ lies within this interval. This condition is sufficient but not necessary. Applying nonautonomous bifurcation methods and imposing more restrictive conditions on the variation of $I(t)$ makes it possible to 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, the case of a periodic input that varies on a slow timescale is analyzed using fast-slow system methods to rigorously establish either a uniformly stable or a bistable response.