合作布尔系统和其他离散系统的Smale和Hirsch定理的类比
Analogues of the Smale and Hirsch Theorems for Cooperative Boolean and Other Discrete Systems
摘要：
定义在状态空间{0，1，.，p-1}^n已用于多种应用，最近用于基因和蛋白质网络的建模。在本文中，我们研究在何种程度上著名的定理Smale和Hirsch，形成（连续）单调动力系统理论的一部分，推广或未能做到这一点的离散情况下。我们表明，任意m维系统不一定嵌入到n维合作系统n=m+1，在Smale定理的连续情况下，但我们表明，这是可能的n=m+2，只要p是足够大的。我们还证明了一个自然的离散模拟强协同意味着非平凡的周期轨道的长度上的界限，并施加类似于李雅普诺夫稳定的条件对所有吸引子。最后，我们探讨了离散系统的不可约性定义的几个自然候选者。虽然这些概念中的一些意味着一个给定的合作系统的强协同性，并施加更严格的限制的周期轨道的长度比强协同性单独，其他合理的定义允许指数长周期轨道的存在。
英文摘要：
Discrete dynamical systems defined on the state space {0,1,...,p-1}^n have been used in multiple applications, most recently for the modeling of gene and protein networks. In this paper we study to what extent well-known theorems by Smale and Hirsch, which form part of the theory of (continuous) monotone dynamical systems, generalize or fail to do so in the discrete case. We show that that arbitrary m-dimensional systems cannot necessarily be embedded into n-dimensional cooperative systems for n=m+1, as in the Smale theorem for the continuous case, but we show that this is possible for n=m+2 as long as p is sufficiently large. We also prove that a natural discrete analogue of strong cooperativity implies nontrivial bounds on the lengths of periodic orbits and imposes a condition akin to Lyapunov stability on all attractors. Finally, we explore several natural candidates for definitions of irreducibility of a discrete system. While some of these notions imply the strong cooperativity of a given cooperative system and impose even tighter bounds on the lengths of periodic orbits than strong cooperativity alone, other plausible definitions allow the existence of exponentially long periodic orbits.
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