Global convergence for time-periodic systems with negative feedback and applications

TL;DR

This paper establishes conditions under which discrete-time negative feedback systems with periodic forcing globally converge to harmonic solutions, with applications to gene regulatory models.

Contribution

It introduces a new amenable condition for global convergence in time-periodic negative feedback systems, extending to biological gene regulation models.

Findings

Proves global convergence to harmonic solutions under specified conditions.

Demonstrates applicability to gene regulatory models with numerical simulations.

Provides a theoretical framework for analyzing periodic negative feedback systems.

Abstract

For the discrete-time dynamical system generated by the Poincare map T of a time-periodic closed-loop negative feedback system, we present an amenable condition which enables us to obtain the global convergence of the orbits. This yields the global convergence to the harmonic periodic solutions of the corresponding time-periodic systems with negative feedback. Our approach is motivated by embedding the negative feedback system into a larger time-periodic monotone dynamical systems. We further utilize the theoretical results to obtain the global convergence to periodic solutions for the time periodically-forced gene regulatory models. Numerical simulations are exhibited to illustrate the feasibility of our theoretical results for this model.