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

Yi Wang, Wenji Wu, Hui Zhou

TL;DR

This work analyzes global convergence for time-periodic negative-feedback systems via the Poincaré map $T$. By embedding the closed-loop dynamics into a time-periodic monotone extended system, the authors derive an amenable integral condition that guarantees convergence to a unique $ au$-periodic solution $r(t)$, with the omega-limit set reducing to the fixed point $r$. They prove the existence of a unique fixed point in the relevant order interval and show that all trajectories starting in the invariant box converge to $r(t)$; in particular, two symmetric periodic solutions coalesce under the condition, ensuring global convergence. The framework is then specialized to time-periodically forced gene regulatory models, where an explicit condition (H) involving $g$ and the production rates $oldsymbol{eta_i}$ ensures a unique $ au$-periodic orbit, supported by numerical simulations that confirm convergence.

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.

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

TL;DR

This work analyzes global convergence for time-periodic negative-feedback systems via the Poincaré map . By embedding the closed-loop dynamics into a time-periodic monotone extended system, the authors derive an amenable integral condition that guarantees convergence to a unique -periodic solution , with the omega-limit set reducing to the fixed point . They prove the existence of a unique fixed point in the relevant order interval and show that all trajectories starting in the invariant box converge to ; in particular, two symmetric periodic solutions coalesce under the condition, ensuring global convergence. The framework is then specialized to time-periodically forced gene regulatory models, where an explicit condition (H) involving and the production rates ensures a unique -periodic orbit, supported by numerical simulations that confirm convergence.

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.
Paper Structure (4 sections, 7 theorems, 72 equations, 5 figures)

This paper contains 4 sections, 7 theorems, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Proof of the main results
  4. Application to time-periodic gene regulatory models

Key Result

Theorem 2.1

Assume $\mathrm{(\mathbf{A1}) }$-$\mathrm{(\mathbf{A4})}$ hold. Then the Poincaré map $T$ for system f(t,x,h(x)) possesses a unique fixed point $r \in [x_0,y_0]_K$ such that

Figures (5)

  • Figure 1: Orbit convergence: $\tilde{T}^n a_0 \uparrow p$ and $\tilde{T}^n b_0 \downarrow q$ with $\omega_{\tilde{T}} \left([a_0,b_0]_C\right)$ remained in the order interval $[a_0,b_0]_C$.
  • Figure 2: Trajectories of $\psi(t,0,\bar{x})$.
  • Figure 3: Trajectories of $x_1(t)$.
  • Figure 4: Trajectories of $x_2(t)$.
  • Figure 5: Trajectories of $x_3(t)$.

Theorems & Definitions (13)

  • Theorem 2.1
  • Corollary 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 3 more