Dynamical Systems with a Cyclic Sign Variation Diminishing Property

In 1970, Binyamin Schwarz defined and analyzed totally positive differential systems (TPDSs), i.e., linear time-varying systems whose transition matrix is totally positive. He showed that any solution of a TPDS satisfies a sign variation diminishing property with respect to the standard number of sign variations. It has been recently shown that several important results on entrainment [stability] in time-varying [time-invariant] nonlinear tridiagonal cooperative systems follow from the fact that the variational equation associated with these nonlinear systems is a TPDS. Thus, the number of sign variations in the vector of derivatives can be used as an integer-valued Lyapunov function. Here we develop the theory of linear cyclic variation diminishing differential systems (CVDDSs). These are systems whose transition matrix satisfies a variation diminishing property with respect to the cyclic number of sign variations. Thus, the cyclic number of sign variations can be used as an integer-valued Lyapunov function for any vector solution of a CVDDS. We show that several known classes of nonlinear cooperative dynamical systems have a variational equation, which is a CVDDS.