Dynamical Systems with a Cyclic Sign Variation Diminishing Property

Abstract

Several studies analyzed certain nonlinear dynamical systems by showing that the cyclic number of sign variations in the vector of derivatives is an integer-valued Lyapunov function. These results are based on direct analysis of the dynamical equation satisfied by the vector of derivatives, i.e. the variational system. However, it is natural to assume that they follow from the fact that the transition matrix in the variational system satisfies a variation diminishing property (VDP) with respect to the cyclic number of sign variations in a vector. Motivated by this, we develop the theoretical framework of linear time-varying systems whose solution satisfies a VDP with respect to the cyclic number of sign variations. This provides an analogue of the work of Schwarz on totally positive differential systems, i.e. linear time-varying systems whose solution satisfies a VDP with respect to the standard (non-cyclic) number of sign variations.