Revisiting Totally Positive Differential Systems: A Tutorial and New Results

TL;DR

This paper reviews the theory of totally positive and nonnegative matrices, their connection to sign variation diminishing properties, and introduces new results and simplified proofs that extend previous stability analyses of certain dynamical systems.

Contribution

It provides a comprehensive tutorial linking classical and recent results on totally positive systems and introduces new generalizations and simplified proofs for stability analysis.

Findings

Matrices with TN or TP exponentials relate to sign variation properties.

New generalized stability results for nonlinear systems.

Simplified proofs of existing theorems on sign variation and system stability.

Abstract

A matrix is called totally nonnegative (TN) if all its minors are nonnegative, and totally positive (TP) if all its minors are positive. Multiplying a vector by a TN matrix does not increase the number of sign variations in the vector. In a largely forgotten paper, 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. 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 provide a tutorial on these fascinating research topics and show that they are intimately related. This allows to derive generalizations of the results by Smillie (1984) and Smith (1991) while simplifying the proofs. It also opens the door to many new and interesting research directions.