  The framework of mutually coupled oscillators on a network has served as a
convenient tool for investigating the impact of various parameters on the
dynamics of real-world systems. Compared to large networks of oscillators,
minimal networks are more susceptible to changes in coupling parameters, the
number of oscillators, and network topologies. In this study, we systematically
explore the influence of these parameters on the dynamics of a minimal network
comprising Stuart-Landau oscillators coupled with a distance-dependent time
delay. We examine three network topologies: ring, chain, and star.
Specifically, for ring networks, we study the effects of increasing nonlocality
from local to global coupling on the overall dynamics of the system. Our
findings reveal the existence of various synchronized states, including splay
and cluster states, a partially synchronized state such as chimeric
quasiperiodicity, and an oscillation quenching state such as amplitude death in
these networks. Moreover, through an analysis of long-lived transients, we
discover novel amplitude-modulated states within ring networks. Interestingly,
we observe that increasing nonlocality diminishes the influence of the number
of oscillators on the overall behavior in these networks. Furthermore, we note
that chain networks, unlike ring networks, do not exhibit perfect synchrony
among the coupled oscillators. In contrast, star networks demonstrate greater
stability and are unaffected by the number of oscillators within the network.
The insights from this study deepen our understanding of the dynamics of
minimal networks and have implications for various fields, ranging from biology
to engineering.
