Analysis of Nonlinear Tridiagonal Cooperative Systems using Totally Positive Linear Differential Systems

Cooperative tridiagonal dynamical systems appear often in biological and engineering applications. The most important theorem for such systems was arguably one proved by Smillie in 1984, and subsequently refined by other authors. Smillie showed that-under mild technical assumptions-precompact trajectories always converge to equilibria. The key to his proof was the construction of an integer-valued Lyapunov function that certifies that the number of sign variations in the vector of derivatives of states eventually stabilizes. This paper shows how to re-derive Smillie's theorem by appealing to results from Binyamin Schwarz, who analyzed the sign variations in solutions of linear systems whose flows are totally nonnegative or totally positive (meaning that all minors are nonnegative or positive, respectively). The connection is through the variational equation associated to the original system. In addition to connecting two seemingly disparate areas of research, the connection allows one to both simplify proofs and extend the validity of Smillie's Theorem.