Projective Convergence Of Monotone Skew-product Semiflows And Investigation On Circadian Rhythms

In this thesis, we first study the strongly monotone skew-product semiflows with minimal base flow. Under the phase-translation group action invariance condition, we establish the convergence property for pseudo-bounded forward orbits. Then, we ap-ply the results to obtain global convergence of the enzymatic futile cycles, as well as the dynamics of certain reaction-diffusion systems, in time-recurrent structure includ-ing periodicity, almost periodicity and almost automorphy. At last, we investigate the model of molecular mechanism of circadian rhythms proposed by Tyson et al (1999). A sufficient condition to guarantee the circadian rhythms is shown, and the numerical integrations coincide with some experimental results very well.This paper is organized as follows:In the introduction, we first recall the history of monotone dynamical system. Typical methods to study the ω-limit set are introduced. We then talk about the skew-product semiflow theory. After that we lay out our main results, including projective convergence of monotone skew-product semiflow with applications to enzymatic futile cycle and a class of reaction-diffusion equations, and an existence criterion of circadian rhythms.In chapter2, we collect some preliminary materials that will be used later. First, we recall the definition of partial order and the induced topology. We then summa-rize some basic conceptions and basic properties of strongly monotone skew-product semiflows. Finally we give a brief introduction about almost periodic functions.In chapter3, we study the strongly monotone skew-product semiflows with min-imal base flow. We prove that the pseudo-bounded trajectory convergent to a1-cover of base space if the system owns phase-translation group action invariance property. Precisely, first we turn to study an induced non-monotone skew-product system. With a completely new method, we prove that all bounded solutions of the induced system convergent to a unique1-cover of base space. The convergence conclusions are then extended to the original skew-product semiflows. In chapter4, we present two examples as the applications of the results in chap-ter3. In the fist example, the global dynamics is fully established for time-dependent enzymatic futile cycles, which are non-monotone systems, in biological molecular net-works. We further obtain in the second example a convergence property for a certain class of reaction-diffusion equations.In chapter5, we give a deeply investigation on the model of circadian rhythms proposed by Tyson et al (1999). By exploring the dynamics of the system, we give an existence criterion for circadian rhythms. Numerical estimations are then presented to explain some former experimental observations, and some of which coincide with their work perfectly.