Bipartite Synchronization and Convergence Analysis for Network of Harmonic Oscillator Systems With Signed Graph and Time Delay

This paper considers the distributed control problem for a network of harmonic oscillator systems with delayed velocity coupling and signed graph topology containing a directed spanning tree. In sharp contrast to the dynamical behaviors of traditional coupled harmonic oscillators over unsigned graphs, it is found that the considered signed network can exhibit some more interesting behaviors, including bipartite synchronization, stabilization, and convergent behavior. First, when the underlying graph is structurally balanced, some necessary and sufficient conditions in terms of coupling strength and time delay are established to achieve bipartite synchronization in the signed network. Then, under the assumption that the signed graph is structurally unbalanced, it is shown that the network state converges either to the origin or to a nontrivial trajectory. In particular, a novel approach is developed to explicitly determine the converged trajectory based on the left and right eigenvectors associated with the simple zero eigenvalue of the signed graph's Laplacian matrix. Finally, the theoretical analysis is illustrated by some numerical examples.