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

TL;DR

This work analyzes a nonautonomous scalar differential equation arising from the Bonifacio-Lugiato optical model with a concave-convex nonlinearity. By leveraging nonautonomous bifurcation theory, hull constructions, and auxiliary concave-linear/linear-convex reductions, it derives precise conditions under which the system exhibits uniform stability or bistability as the time-varying input $I(t)$ varies. It provides sharp criteria via intervals $\mathcal{I}_1(c)$, $\mathcal{I}_2(c)$, and $\mathcal{I}_3(c)$ (depending on the parameter $c$, with $c>4$ enabling bistability) and develops refined estimates for periodic/almost-periodic inputs, including d-concavity band analyses. Additionally, the paper investigates slow periodic forcing through slow-fast methods to characterize relaxation oscillations versus bistable responses, with implications for optical fluorescence modeling and control in driven cavity systems.

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.

Figures (11)

• Figure 1: Sketch of a ring cavity formed by four mirrors, containing a medium composed of two-level atoms (in blue). The amplitude of the incident field is proportional to the parameter $\lambda$, while the amplitude of the transmitted field is proportional to the variable $x$. A third arrow represents the reflected field, which is not included in the equation under study. The sketch is based on Fig. 1 from Ref. bonifaciolugiato3.
• Figure 2: Bifurcation diagram for the autonomous model \ref{['eq:autonomousbonifacio']} for $x\geq0$ and any fixed $c>4$: in this drawing, $c=5$. The hyperbolic attractive fixed points are represented in red, while the hyperbolic repulsive fixed points are shown in blue. The non-hyperbolic fixed points, which correspond to bifurcation points, are depicted in black.
• Figure 3: Representation of the functions $h_1(c)$, $h_2(c)$ and $h_3(c)$ given by \ref{['def:h1']}, \ref{['def:h2']} and \ref{['def:h3']} on $[4,10]$.
• Figure 4: Representation of $\mathfrak{d}\mapsto\mathfrak{d}/\sqrt{\mathfrak{d}^2+\theta^2}$ for different values of $\theta$.
• Figure 5: Numerical depiction of the $2\pi$-periodic function $y(t)=\sum_{n=1}^{51}(\cos(n\,t)-\sin(n\,t))$ on $[\pi/2,5\pi/2]$.

Theorems & Definitions (32)

• Definition 2.1
• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Lemma 4.1
• proof
• Remark 4.2
• Lemma 4.3