Time-dependent invariant sets in system dynamics

The paper provides new results on the flow (positive) invariance of the families of 0-symmetrical sets, which are defined by arbitrary Hölder norms and time-dependent diagonal matrices. Thus, we introduce the concept of “diagonal invariance” as a system property with local or global character. For this property, we formulate sufficient conditions in the case of time-variant or -invariant, nonlinear systems and necessary and sufficient conditions in the case of time-variant or -invariant, linear systems. We also derive a comparison method that allows exploring the diagonal invariance of time-variant or -invariant, nonlinear or linear systems, by using time-invariant, linear comparison systems. We illustrate the applicability of the diagonal invariance criteria to the important class of nonlinear systems described by Hopfield neural networks. These new results represent a meaningful generalization of some previous researches developed by the same authors.