Asymptotically 2-Contractive Systems: Converging properties and Applications

In this paper we show that the positive limit sets of asymptotically 2-contractive systems, i.e., systems with a vector field that converges uniformly to a 2-contractive vector field, consist of isolated equilibria. This result is established by combining a known result about asymptotically autonomous systems with a characterization of the chain recurrent sets of 2-contractive systems with hyperbolic equilibria. We apply this result to characterize the limit sets of cascades of 2-contractive systems, synchronizing networks, and systems with converging inputs.