The spark of synchronization in heterogeneous networks of chaotic maps

Abstract: We investigate the emergence of synchronization in heterogeneous networks of chaotic maps. Our findings reveal that a small cluster of highly connected maps is responsible for triggering the spark of synchronization. After the spark, the synchronized cluster grows in size and progressively moves to less connected maps, eventually reaching a cluster that may remain synchronized over time. We explore how the shape of the network's degree distribution affects the onset of synchronization and derive an expression based on the network's construction that determines the expected time for a network to synchronize. Understanding how the network structure affects the spark of synchronization is particularly important for the control and design of more robust systems that require some level of coherence between a subset of units for better functioning. Numerical simulations in finite-sized networks are consistent with this analysis.