Modulus synchronization in a network of nonlinear systems with antagonistic interactions and switching topologies

• The concept of modulus synchronization is introduced for nonlinear networks with antagonistic interactions.
• We use matrix measure and contraction theory.
• Modulus synchronization can be evaluated by the time average of some matrix measures.

This paper studies the collective behavior in a network of nonlinear systems with antagonistic interactions and switching topologies. The concept of modulus synchronization is introduced to characterize the case that the moduli of corresponding components of the agent (node) states reach a synchronization. The network topologies are modeled by a set of directed signed graphs. When all directed signed graphs are structurally balanced and the nonlinear system satisfies a one-sided Lipschitz condition, by using matrix measure and contraction theory, we show that modulus synchronization can be evaluated by the time average of some matrix measures. These matrices are about the second smallest eigenvalue of undirected graphs corresponding to directed signed graphs. Finally, we present two numerical examples to illustrate the effectiveness of the obtained results.