Synchronization Analysis on Two-Layer Networks of Fractional-Order Systems: IntraIayer and InterIayer Synchronization

The paper focuses on a two-layer network composed of fractional-order oscillators, and investigates the mesoscale collective dynamical behaviors of the network, i.e., intra-layer synchronization and inter-layer synchronization. Specifically, the necessary constraint conditions are firstly presented for achieving intra-layer synchronization and inter-layer synchronization. With the constraints, based on the Lyapunov stability theory and the fractional-order system theory, sufficient conditions for inter-layer synchronization and intra-layer synchronization are obtained respectively. Interestingly, the conditions cover the basic synchronization criteria bridging the relationship of interacting between the topologies, inner coupling functions and coupling strengthens of the intra-layer and the inter-layer. Finally, numerical examples are provided to further verify the theoretical results.