Methodology for a nullcline-based model from direct experiments: Applications to electrochemical reaction models

A method is proposed for the construction of simple yet accurate mathematical models for description of dynamics of nonlinear systems based on the concept of nullclines. The nullclines are functional relationships between the essential variables of a dynamical system where the rate of change of a species is constrained to zero. When the time scales of the essential variables are separated the dynamics can often be described by motion along nullclines of the fast variable. We propose a methodology with which the nullclines can be obtained from direct control experiments. The goal is achieved by a combination of an adaptive and proportional controller acting on the fast and the slow variables of the system. It is demonstrated in the numerical simulations of two- and three-variable electrochemical models that the nullclines can be extracted from control of experimentally feasible control parameters (circuit potential and rotation rate). As an extension of the methodology, we propose a nullcline based technique which is capable of reproducing the temporal behavior of the system’s variables (e.g., waveform of the oscillations). The numerical simulations indicate that the nullcline based model information from direct experiments can effectively predict the system’s dynamics and thus the technique holds promise for an alternative modeling route to traditional kinetics/mass transfer based approaches.