Inter-layer synchronization of periodic solutions in two coupled rings with time delay

Highlights: • A coupled rings model is developed to avoid the weakness of a ring topology. • A divide and conquer algorithm is used to give distributions of the eigenvalues. • Stability of the periodic solutions is analyzed when system has repeated eigenvalues. • The inter-layer synchronization in the coupled rings is analyzed. • The proposed model can be used to achieve the "back and recovery" procedure. -- Abstract: A coupled two-ring-network with time delay is developed to avoid the weakness of a simple ring topology. The divide and conquer algorithm is used to analyze the distribution of the eigenvalues of the adjacent matrix. The stability of the equilibrium and inter-layer synchronization of the periodic dynamics are studied, and the dynamical interactions between coupled rings which lead to complicated mirror and reflection waves are further analyzed. Special attention is paid on the generalized synchronization between two rings when the adjacent matrix has repeated eigenvalues. Numerical simulations verify the theoretical results, and show that the appropriate settings of the time delay and transverse coupling can guarantee the "backup and recovery" procedure.