  We take into consideration the evolution of particle size in a monodisperse
aerosol population during activation and deactivation of cloud condensation
nuclei (CCN). The phase portrait of the system derived through a
weakly-nonlinear analysis reveals a saddle-node bifurcation and a cusp
catastrophe. An analytical estimate of the activation timescale is derived
through estimation of the time spent in the saddle-node bifurcation bottleneck.
Numerical integration of the system portrays two types of
activation/deactivation hystereses: one associated with the kinetic limitations
on droplet growth when the system is far from equilibrium, and one occurring
close to equilibrium and associated with the cusp catastrophe. The hysteretic
behaviour close to equilibrium imposes stringent time-resolution constraints on
numerical integration, particularly during deactivation.
