Behavior of Totally Positive Differential Systems Near a Periodic Solution

TL;DR

This paper analyzes the behavior of totally positive differential systems near periodic solutions, revealing how trajectories converge or diverge, with applications demonstrated in a biological model.

Contribution

It applies spectral theory of oscillatory matrices to understand the local dynamics of TPDS near periodic solutions, a novel approach in this context.

Findings

Trajectories converge to periodic solutions in periodic TPDS.

Spectral analysis identifies fastest and slowest convergence directions.

Application to a biological model illustrates practical relevance.

Abstract

A time-varying nonlinear dynamical system is called a totally positive differential system (TPDS) if its Jacobian admits a special sign pattern: it is tri-diagonal with positive entries on the super- and sub-diagonals. If the vector field of a TPDS is T-periodic then every bounded trajectory converges to a T-periodic solution. In particular, when the vector field is time-invariant every bounded trajectory of a TPDS converges to an equlbrium. Here, we use the spectral theory of oscillatory matrices to analyze the behavior near a periodic solution of a TPDS. This yields information on the perturbation directions that lead to the fastest and slowest convergence to or divergence from the periodic solution. We demonstrate the theoretical results using a model from systems biology called the ribosome flow model.