Quantification of synchronization

Abstract

Synchronization is a universal behavior that is crucial in a wide array of biological, physical, and technical systems. Some examples are flashing fireflies, cardiac and neural processes, non-linear electronic circuits, coupled lasers, and plasma. The occurrence of self-organized synchrony in nature may appear implausible at first. However, this phenomenon emerges with the development of an interaction channel, which is known as coupling. Since the work of synchronization of chaotic systems started, several types of synchronization have been described theoretically and observed experimentally. Great progress has been made in the field, but the boundaries between the synchronization states are often blurred, and such boundary regimes have received less attention in the past. Observing the subtle details between different degrees of synchronization is a challenging task. In the literature, such issues have not been discussed systematically up to now.

This dissertation work is initiated with the objective to fill some of those gaps by applying a quantitative Bayesian statistical approach. The unified approach presented here will enable us to quantify the synchronization of chaotic systems by accurately identifying the coupling strength from the available measured data. The method is sensitive with respect to changes in the measured time series as well as to the underlying geometry of the attractor. It will allow us to distinguish specific types of synchronization by continuously monitoring degrees of synchronization, for any values of the coupling strength parameter. We initiated this study with small-dimensional chaotic system analyses and extended the approach to a large-scale first-order Kuramoto network. We further advance our analysis from identical Kuramoto network coupling to complex Kuramoto networks with different coupling parameters and directions.

The statistical analysis of several widely known coupled systems will permit us to distinguish and quantify well-known and less studied synchronization states. In this work, we discuss fragile states such as emerging synchronization, chaos suppression, escape events, or a quasi complete synchronization, all occurring at different couplings strengths of the respective systems. Some of these states are studied for the first time in this work.