Improving Reduced Variable Models for Complex Systems via Experiment

Populations of simple interacting oscillators can give rise to complex dynamics. Real-world examples of such systems abound in biology, chemistry, and physics. Reduced-variable models can describe these systems. The phase model is a particularly successful reduced-variable model which, in its simplest form, each element is represented by one variable, the phase of oscillation. The phase model can be constructed from observable variables, without specific knowledge of underlying phenomenological behavior. Such reduced variable models are easier to construct and more generalizable than models derived from underlying physics or chemistry. In this dissertation, we study the dynamics of a population of coupled electrochemical oscillators in order to modify the phase model to expand its applicability. Specifically, we develop a two-phase model, a phase and radius model, and models with network coupling.

We analyze data from two coupled oscillators with a standard one-dimensional phase model and a newer two-dimensional phase model. The same quantity of data is needed for either model. The two-dimensional analysis reveals behaviors and coupling parameters in theoretical and experimental examples that the one- dimensional analysis does not show. The two-dimensional model could be useful in systems that exhibit learning due to its ability to distinguish stimulation and reaction.

We examine clustering on small networks with time-delayed interactions. Our experimental results agree with numerical and analytical results from a phase and amplitude model (the Stuart-Landau model). For near-sinusoidal oscillators, the standard Stuart-Landau model successfully describes clustering dynamics. We extend the Stuart-Landau model to describe clustering dynamics for higher harmonic oscillators. We present examples of asymmetrical clusters arising from networks of higher harmonic oscillators. We show that the amplitude of oscillation is functionally influenced by coupling; we believe this ”amplitude coupling function” has not been previously described. This function can be constructed from the original time series with no additional measurements.

Recently, combined phase and amplitude models have gained attention; we explore the conditions where a phase and amplitude model is useful. We show experimentally a change in cluster state due only to changes in coupling strength. Such a transition is not possible with a phase-only model. Simulations with the extended Stuart-Landau model match experimental results. We demonstrate a method for predicting the amplitude coupling function from the coupling. Prior to this study, the relationship between stimulation and response was not known for the amplitude. We suggest that the phase and amplitude model will be most useful (1) in modeling high-harmonic oscillations, and (2) where coupling exceeds the ”weak coupling” approximation of the phase-only model.