Boundedness of Prime Periods of Stable Cycles and Convergence to Fixed Points in Discrete Monotone Dynamical Systems

Source abstract

In this paper the boundedness of minimal periods of linearly stable cycles for discrete, strongly order-preserving semigroups $(F_0^n )_{n \in N}$ in bounded subsets of an ordered Banach space is proved. It is further shown that this bound is not increased by small perturbations of $F_0$. Of particular interest is the case where the only linearly stable cycles of $F_0$ are fixed points. Employing a recent result of Poláčik and Tereščák, the typical convergence of relatively compact orbits and for perturbed systems then follow. The results are applied to classes of time-periodic reaction-diffusion equations and give typical convergence to periodic solutions.