Phase-translation group actions on strongly monotone skew-product semiflows (Q2839955)

This paper studies strongly monotone skew-product semiflows with invariant phase-translation group actions. The semiflow is of the form NEWLINE\[NEWLINE \Pi:X\times Y \times \mathbb{R}^+ \rightarrow X \times Y,\quad (x,g,t) \mapsto \Pi_t(x,g) = (\phi_t(x,g),g\cdot t) NEWLINE\]NEWLINE with a minimal compact metric base \((Y,\mathbb{R})\). The space \(X\) is a strongly ordered Banach space with normal cone \(X^+\) such that \(\Pi\) is strongly monotone and completely continuous (for any bounded set \(E\subset X\), the sets \(\Pi_t(E\times Y)\) are relatively compact). For a fixed \(v\in \mathrm{Int}X_+\) with \(\|v\|=1\), the phase-translation group \(G\) consists of the translations \(a:X\rightarrow X\), \(a\cdot x = x + av\), \(a\in\mathbb{R}\). It is assumed that \(G\) commutes with \(\Pi\) in the sense that \(\phi_t(a\cdot x,g) = a \cdot \phi_t(x,g)\) for all \((x,g)\in X\times Y\), \(t\geq0\), \(a\in G\). A basic tool used to study the orbit structure of \(\Pi\) is the projection \(\pi:X\rightarrow X_0\) onto the null space \(X_0\) of a bounded linear functional \(f\) on \(X\) with \(fv = 1\). Then one obtains an induced skew-product semiflow \(\tilde{\Pi}_t\) on \(X_0 \times Y\) by projection via \(\pi\). A compact \(\Pi\)-invariant set \(E\subset X\times Y\) is called a \(1\)-cover of \(Y\) based on \(\Pi\) if \(E\) intersects each \(Y\)-fiber in exactly one point. In this setup, the authors prove two results:NEWLINENEWLINE{ Theorem A:} If \((\tilde{x}_1,g),(\tilde{x}_2,g)\in X_0\times Y\) have bounded \(\tilde{\Pi}\)-forward orbits, then NEWLINE\[NEWLINE \|\tilde{\phi}_t(\tilde{x}_1,g) - \tilde{\phi}_t(\tilde{x}_2,g)\| \rightarrow 0 \mathrm{\quad for } t\rightarrow\infty. NEWLINE\]NEWLINE In particular, there exists at most one \(1\)-cover of \(Y\) based on \(\tilde{\Pi}\) and for every bounded forward orbit \(\tilde{\mathcal{O}}^+(\tilde{x},g)\) the \(\omega\)-limit set \(\tilde{\omega}(\tilde{x},g)\) is a \(1\)-cover of \(Y\) based on \(\tilde{\Pi}\).NEWLINENEWLINE{ Theorem B:} Let \(\mathcal{O}^+(x,g)\) be a forward orbit of \(\Pi\). Then (1) the \(\omega\)-limit set \(\omega(x,g)\) is a \(1\)-cover of \(Y\) if \(\mathcal{O}^+(x,g)\) is bounded, and (2) the \(\omega\)-limit set \(\tilde{\omega}(\pi x,g)\) is a \(1\)-cover of \(Y\) if the projection of the forward orbit \(\mathcal{O}^+(x,g)\) to \(X_0 \times Y\) is bounded.NEWLINENEWLINEBased on these results, the authors prove global convergence of certain chemical reaction networks, described by nonautonomous ODEs with time-recurrent right-hand sides. After a coordinate change to so-called reaction coordinates, a standard construction, using the hull of the right-hand side as the base space, yields a skew-product semiflow with minimal base flow. Under the assumption that this semiflow is strongly monotone, it is shown that every solution converges to an almost-periodic, almost-automorphic, periodic or quasi-periodic solution, depending on the type of time-dependence of the right-hand side. In particular, the authors show that this result can be applied to a simple phosphorylation/dephosphorylation process. Finally, an application to reaction-diffusion systems is presented.