Convergence and structure theorems for order-preserving dynamical systems with mass conservation
Convergence and structure theorems for order-preserving dynamical systems with mass conservation
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DOI：
10.3934/dcds.2020129
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
T. Ogiwara;D. Hilhorst;H. Matano
中科院分区：
文献类型：
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作者：
T. Ogiwara;D. Hilhorst;H. Matano
我们建立了一个一般理论的存在性的不动点和收敛的轨道在保序半动力系统具有一定的质量守恒性质（或等价地，第一积分）。基空间是一个有序度量空间，我们不假设系统的可微性，也不要求基空间中的线性结构。我们的第一个主要结果指出，任何轨道要么收敛到一个不动点或逃逸到无穷大（收敛定理）。这将在不假设存在不动点的情况下得到证明。我们的第二个主要结果指出，一个不动点的存在意味着存在一个全序不动点的连续统（结构定理）。后一个结果，当应用到一个线性问题，其中\开始{document}$0 $\end{document}总是一个不动点，自动意味着存在正不动点。我们的结果扩展了Arino（1991），Mierczynski（1987）和Banaji-Angeli（2010）的相关工作，证明非常简单。我们将我们的结果应用于许多问题，包括具有时间周期或自治系数的分子电机模型、某些类别的反应扩散系统和延迟微分方程。
We establish a general theory on the existence of fixed points and the convergence of orbits in order-preserving semi-dynamical systems having a certain mass conservation property (or, equivalently, a first integral). The base space is an ordered metric space and we do not assume differentiability of the system nor do we even require linear structure in the base space. Our first main result states that any orbit either converges to a fixed point or escapes to infinity (convergence theorem). This will be shown without assuming the existence of a fixed point. Our second main result states that the existence of one fixed point implies the existence of a continuum of fixed points that are totally ordered (structure theorem). This latter result, when applied to a linear problem for which \begin{document}$ 0 $\end{document} is always a fixed point, automatically implies the existence of positive fixed points. Our result extends the earlier related works by Arino (1991), Mierczynski (1987) and Banaji-Angeli (2010) considerably with exceedingly simpler proofs. We apply our results to a number of problems including molecular motor models with time-periodic or autonomous coefficients, certain classes of reaction-diffusion systems and delay-differential equations.