Translation-invariant monotone systems II: Almost periodic/automorphic case

Abstract

Abstract. This paper studies almost periodic/automorphic monotone systems with positive translation invariance via skew-product flows. It is proved that every bounded solution of such systems is asymptotically almost periodic/automorphic. Applications are made to a chemical reaction network, especially to enzymatic futile cycles with almost periodic/automorphic reaction coefficients.

Poisson Stable Motions and Global Attractors of Symmetric Monotone Nonautonomous Dynamical Systems

“…In present work we study a class of monotone nonautonomous dynamical systems with symmetry. The writing of this article was motivated by works D. Angeli and E. Sontag [1,2], D. Angeli, P. Leenheer and E. Sontag [3] (for autonomous systems), H. Hu and J. Jiang [22,23] (for periodic and almost periodic systems) and Q. Liu and Y. Wang [28] (for almost periodic and almost automorphic systems). We study these problems within the framework of general non-autonomous dynamical systems (cocycles).…”

Phase-translation group actions on strongly monotone skew-product semiflows

“…Although the new system in reaction coordinates has been known to be strongly monotone, a careful examination immediately yields that the change to such a new system does not seem particularly useful because there is no guarantee that the solutions of the new system are bounded. (Recently, Hu and Jiang [21,22] discussed such a new monotone system under the assumption of boundedness for every solution.) In fact, as pointed out by Angeli et al in [3, p. 596]: "this issue constitutes the main technical difficulty that needs to be surmounted in order for us to obtain the convergence results for the system".…”