  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.
