Nonlinearities Shape the Response Patterns to Oscillatory Inputs in a Caricature Electrochemical Cell Model

We investigate the nonlinear mechanisms of generation of preferred frequency response patterns to oscillatory inputs in a phenomenological (caricature) model that captures the type of nonlinearities present in electrochemical (EC) cells and is simple enough to begin to develop the dynamical systems tools necessary for the understanding of the relationship between the model nonlinearities and the response patterns. Previous work has shown that linearized EC models exhibit resonance (preferred frequency response to oscillatory inputs at a nonzero, resonant, input frequency) in parameter regimes where the stable equilibrium is either a node (no intrinsic oscillations) or a focus (damped oscillations). We use a combination of numerical simulations and dynamical systems tools to understand how the model nonlinearities, partially captured by the geometry of the nullclines in the phase-space diagram, shape the response patterns to oscillatory inputs away from the validity of the corresponding linearization. We develop and adapt an extended version of the classical phase-plane diagram that allows us to track the evolution of the response trajectories and their interaction with the so-called moving nullclines (in response to the oscillatory inputs). We use this approach to explain the mechanisms of generation of nonlinear resonant patterns, the nonlinear amplification/attenuation of these patterns by increasing input amplitudes, and the mechanisms of generation of more complex patterns of mixed-mode-type, where the stationary amplitude response is not uniform across cycles.