Identification of models described by two differential equations from one scalar time series

Reconstruction of any model from time series is the most straightforward way to its identification and validation. Unfortunately, for most known mathematical models of physical and biological systems, reconstruction cannot be done by means of existing approaches due to the lack of data: not all necessary variables can be measured, and unmeasurable (hidden) variables cannot be achieved by means of numerical integration, differentiation or time shift. The special hidden variable approaches are very fragile and can solve the problem only partly: they provide possibility to reconstruct some parameters if the equations are completely known. Here, the new approach is proposed for a certain large class of models described by two differential equations. This class includes many popular models from different areas of science, including Lotka--Volterra population dynamics equation, reduced model of p53-protein concentration dynamics and a few neuron models: FitzHugh--Nagumo, Wilson--Cowan and Morris--Lecar. The approach demands measuring only one variable from two dynamical variables of the system. It is based on the identification of equations integrated in time, being much more stable against measurement noise than most known approaches. Also, it does not use an explicit approximation for some nonlinearities, providing more robust results.