Flow-Invariance and Stability Analysis for a Class of Nonlinear Systems with Slope Conditions

摘要

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 sets approaching the equilibrium point allow a componentwise exploration of the stability properties, offering additional information with respect to the traditional framework (that expresses a global knowledge, built in terms of norms). Within this context, we are able to point out some important dynamical aspects that remained hidden for other works relying only on the standard tools of stability analysis. The refinement induced by the componentwise point of view is also revealed by two numerical examples.