Determining the Critical Point of a Sigmoidal Curve via its Fourier Transform

Abstract

A sigmoidal curve y(t) is a monotone increasing curve such that all derivatives vanish at infinity. Let t(n) be the point where the nth derivative of y(t) reaches its global extremum. In the previous work on sol-gel transition modelled by the Susceptible-Infected- Recovered (SIR) system, we observed that the sequence {t(n)} seemed to converge to a point that agrees qualitatively with the location of the gel point [2]. In the present work we outline a proof that for sigmoidal curves satisfying fairly general assumptions on their Fourier transform, the sequence {t(n)} is convergent and we call it "the critical point of the sigmoidal curve". In the context of phase transitions, the limit point is interpreted as a junction point of two different regimes where all derivatives undergo their highest rate of change