New results of qualitative analysis are presented for a class of dynamical systems (including the Hopfield neural networks) whose nonlinearities satisfy certain slope conditions. The main instrument of this analysis consists in the individual monitoring of the state-trajectories by considering time-dependent rectangular sets that are forward invariant with respect to the dynamics of the investigated systems. Particular requirements for the rectangular

The componentwise stability is a special type of asymptotic stability which ensures the individual monitoring of each state-space variable of a dynamical system. For an interval Hopfield neural network (IHNN), sufficient conditions are provided to analyze the absolute componentwise stability with respect to a class of activation functions (CAF). Both continuous- and discrete-time dynamics are considered. The conditions are formulated in terms of