Revisiting totally positive differential systems: A tutorial and new results

A matrix is called totally nonnegative (TN) [totally positive (TP)] if all its minors are nonnegative [positive]. Multiplying a vector by a TN matrix does not increase the number of sign variations in the vector. In a largely forgotten paper, Binyamin Schwarz (1970) considered matrices whose exponentials are TN or TP. He also analyzed the evolution of the number of sign changes in the vector solutions of the corresponding linear system. His work, however, considered only linear systems. In a seemingly different line of research, Smillie (1984), Smith (1991), and others analyzed the stability of nonlinear tridiagonal cooperative systems by using the number of sign variations in the derivative vector as an integer-valued Lyapunov function. We show that these two research topics are intimately related. This allows to derive important generalizations of the results by Smillie (1984) and Smith (1991) while simplifying the proofs. These generalizations are particularly relevant in the context of control systems. Also, the results by Smillie and Smith provide sufficient conditions for analyzing stability based on the number of sign changes in the vector of derivatives and the connection to the work of Schwarz allows to show in what sense these results are also necessary. We describe several new and interesting research directions arising from this new connection.