Analogues of the Smale and Hirsch theorems for cooperative Boolean and other discrete systems

作者：Enciso German A^*; Just Winfried

来源：Journal of Difference Equations and Applications, 2012, 18(2): 223-238.

DOI：10.1080/10236190903357535

• Boolean networks
• monotone systems
• periodic solutions
• mathematical biology

Discrete dynamical systems defined on the state space Pi = {0, 1,..., p - 1}(n) have been used in multiple applications, most recently for the modelling 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. %26lt;br%26gt;We show 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. %26lt;br%26gt;We also prove that strict cooperativity, a natural weakening of the notion of strong cooperativity, implies non-trivial bounds on the lengths of periodic orbits in discrete systems 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 strict cooperativity of a given cooperative system and impose even tighter bounds on the lengths of periodic orbits than strict cooperativity alone, other plausible definitions allow the existence of exponentially long periodic orbits.

• 出版日期2012