On the possibility of explosive synchronization in small world networks

The purpose of this work is to study the possibility of the existence of explosive chaotic synchronization phenomenon in small world networks of nonlinear oscillators in the same way as it has been observed for complex networks of nonlinear elements with a random or scale-free topologies of links between network nodes. Methods. In this work, along with numerical modeling, an analytical approach has been used to describe the behavior of network nonlinear elements below the threshold of the occurrence of a completely synchronous state of all network oscillators. Results. It has been shown that in networks of nonlinear oscillators with topologies of links between nodes being the "ring" or "small world" type, with an increase in the coupling strength, an abrupt transition to the completely synchronous dynamics of all network oscillators is possible, just as it happens within the random and scale-free networks where the explosive synchronization phenomenon has been discovered and studied. With the help of theoretical consideration, a mechanism leading to the emergence of this abrupt transition to a synchronous state in ring and small-world networks has been revealed. This mechanism is associated with the formation of two independent synchronous clusters coexisting within network. Conclusion. The paper considers the mechanisms leading to a sharp "explosive" transition to completely synchronous dynamics of all elements in ring and small world networks of nonlinear oscillators, analytical relations are obtained that describe this transition and make it possible to explain the revealed phenomenon.