Modern scientometrics borders with chemometrics and bibliographical statistics [1]. Scientometricists measure the number of articles and construct time series. However, mathematical apparatus of time series can not explain attractors and bifurcations that have place in trends of analytical instruments.

Figure shows dynamics of yearly number of publications in the journal Analytical Chemistry published by American Chemical Society. One can easily see stages of increase, decrease and new increase that we named as the stage of "second birth".

The phenomenon of "second birth" can be explained by equation, which based on classic logistic growth model [2]. The main modification is connected with including basic parameters as linear decreasing functions, i.e. variant values. This equation is presented as dy/dt = k₀(1 – at)y(N₀(1 – bt) – y); y(0) = n , where n number of publications (articles) at the first year after birding colorimetry as scientific direction, t — time, a, k₀, N₀ and b — basic parameters of mathematic model. The estimation of this parameters values was the general problem. We used traditional numerical iterations method (co-ordinate or spiral descent) for searching optimum of quality functional. But search of point of first iteration is based on solution the system of four single (but unlinear) equations. For comparison, we present the first iteration of parameters values a^* = 0.122; b^* = 0.022; N₀^* = 97.1; k₀^* = 0.0134 and the final solution: a = 0.1280, b = 0.02242, N₀ = 120.0, k₀ = 0.0020

Another examples of the "second birth" stage in analytical instrumentation trends are time-of-flight mass spectrometers, paper chromatography and, perhaps, potentiometry. But the most types of analytical instruments don't have the stage of "second birth" (or "death of scientific direction").

References:
1. Malinetskii G.G., Kurdyumov S.P. Nonlinear Dynamics and Prediction Problems // Herald of RAS. 2001. V. 71, No. 3. P. 210–224.
2. Arkhipov D.B., Bulianitsa A.L., Scherbakov A.P. Simulation of Trends of Analytical Insrtumentation Based o Logistic Equation with Coefficients Depending on Time // 3-d All-Russian Conference "Analytical Instruments". St. Petersburg. 2008. P. 20–21.