Inter-layer synchronization on a two-layer network of unified chaotic systems: The role of network nodal dynamics.

The interplay between synchronization across layers and structures within layers still remains far unclear in multi-layer networks. Via our newly-constructed two-layer network consisting of the unified chaotic systems where the link within layer is linearly coupled and the link across layers is formed via sharing a common signal, this paper investigates the inter-layer synchronization on two-layer networks from the perspective of network nodal dynamics, as well as explore the influence of intra-layer network structures on inter-layer synchronization. Here we derive an analytical sufficient condition for inter-layer synchronization, which is only dependent on network nodal dynamics, showing a handful of Lorenz-type oscillators including the classic one are able to successfully realize inter-layer synchronization of the network for any connected intra-layer topological structures and diagonal coupling matrices. Further, four regular networks, small-world, scale-free and random networks are respectively used to run numerical simulations. Numerical results verify the presented theoretical results, and also show that Lü-type as well as Chen-type oscillators networks fail to realize inter-layer synchronization no matter which kinds of topologies and coupling matrices within layers. Specifically, for the case of identical topologies within layers, the chain-type and ring-type structures can stimulate larger range of Lorenz-type oscillators into synchronization across layers under the assumption of diagonal coupling matrices, but the fully-connected and star-type structures fail, and the non-diagonal intra-layer coupling matrix on the whole weaken the inter-layer synchronizability. For the case of nonidentical topologies within layers, the connection structure within layers can hardly enlarge the range of oscillators for synchronization across layers, even narrow the range, but some structures may enlarge the range of coupling strengthes for inter-layer synchronization. • Synchronization is studied from the perspective of network nodal dynamics. • A new two-layer network model is constructed. • An analytical sufficient condition for inter-layer synchronization is presented. • Investigations on the range of oscillators for synchronization are provided. [ABSTRACT FROM AUTHOR]