The paper explores the flow (positively) invariant sets with respect to the trajectories of recurrent neural networks (RNNs). There are considered two types of sets, namely with arbitrary time-dependence and exponentially decreasing. The sets can have general shapes, defined by Hölder
-norms. The first part of the paper develops criteria for testing the existence of invariant sets. The second part analyzes the connections between the invariant sets and the stability of the RNN equilibrium points. Besides the novelty and the theoretical interest of the whole approach, the results corresponding to the usual p-norms (
= 1,2, ∞ ) yield numerically tractable procedures for testing the invariance properties.