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

Abstract For the infinite dimensional system combining time-delayed negative feedback and time-periodic driving, we investigate the discrete-time dynamical system generated by the Poincaré map T of this system and 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 delayed systems with negative feedback. Our approach is motivated by embedding the negative feedback system into a larger time-periodic monotone delayed dynamical systems. We further utilize the theoretical results to obtain the global convergence to periodic solutions for the time periodically-forced gene regulatory models with distribution delay. Numerical simulations are exhibited to illustrate the feasibility of our theoretical results for these models.