Matrix framework for analysis of invariant sets

For a large class of nonlinear dynamical systems, we provide results that allow testing the existence of invariant sets with respect to the state space trajectories. These sets are considered as depending on time, in order to ensure the generality of the approach. The results operate as sufficient conditions formulated in terms of matrix measures associated with induced matrix norms. For linear dynamical systems, the derived conditions are also necessary and they incorporate, as particular cases, characterizations of flow in variance given by previous works, which exploited a different background. The applicability of our results is illustrated for interval dynamical systems and recurrent neural networks. A separate discussion focuses on special types of invariant sets and their links with the stability properties.