IEEE Transactions on Automatic Control, Vol.65, No.3, 941-954, 2020
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 & x00A0;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 & x00A0;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 & x00A0;(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 & x00A0;CVDDS. We show that several known classes of nonlinear cooperative dynamical systems have a variational equation, which is a & x00A0;CVDDS.
Keywords:Nonlinear systems;Lyapunov methods;Time-varying systems;Stability analysis;Standards;Cooperative systems;Indexes;Compound matrices;cooperative dynamical systems;cyclic sign variation diminishing property (VDP);minor;stability analysis;totally positive (TP) matrices;TP differential systems (TPDS)