How clock heterogeneity affects synchronization and can enhance stability

Abstract: The production process of integrated electronic circuitry inherently leads to large heterogeneities on the component level. For electronic clock networks this implies detuned intrinsic frequencies and differences in coupling strength and the characteristic time-delays associated with signal transmission, processing and feedback. Using a phase-model description, we study the effects of such component heterogeneity on the dynamical properties of synchronization in networks of mutually delay-coupled Kuramoto oscillators. We test the theory against experimental results and circuit-level simulations in a prototype system of mutually delay-coupled electronic clocks, so called phase-locked loops. Contrary to the hindering effects of component heterogeneity for the synchronization in hierarchical networks, we show that clock heterogeneities can enhance self-organized synchronization in networks with flat hierarchy. That means that beyond the optimizations that can be achieved by tuning homogeneous coupling strengths, time-delays and loop-filter cut-off frequencies, heterogeneities in these system parameters enable much better optimization of perturbation decay rates, the stabilization of synchronous states and the tuning of phase-differences between the clocks. Our theory enables the design of custom-fit synchronization layers according to the specific requirements and properties of electronic systems, such as operational frequencies, phase-relations and e.g. transmission-delays. These results are not restricted to electronic systems, as signal transmission, processing and feedback delays are common to networks of spatially distributed and coupled autonomous oscillators.