Sharpened dynamics alternative and its $C^1$-robustness for strongly monotone discrete dynamical systems

For strongly monotone dynamical systems, the dynamics alternative for smooth discrete-time systems turns out to be a perfect analogy of the celebrated Hirsch's limit-set dichotomy for continuous-time semiflows. In this paper, we first present a sharpened dynamics alternative for $C^1$-smooth strongly monotone discrete-time dissipative system $\{F_0^n\}_{n\in \mathbb{N}}$ (with an attractor $A$), which concludes that there is a positive integer $m$ such that any orbit is either manifestly unstable; or asymptotic to a linearly stable cycle whose minimal period is bounded by $m$. Furthermore, we show the $C^1$-robustness of the sharpened dynamics alternative, that is, for any $C^1$-perturbed system $\{F_\epsilon^n\}_{n\in \mathbb{N}}$ ($F_\epsilon$ not necessarily monotone), any orbit initiated nearby $A$ will admit the sharpened dynamics alternative with the same $m$. The improved generic convergence to cycles for the $C^1$-system $\{F_0^n\}_{n\in \mathbb{N}}$, as well as for the perturbed system $\{F_\epsilon^n\}_{n\in \mathbb{N}}$, is thus obtained as by-products of the sharpened dynamics alternative and its $C^1$-robustness. The results are applied to nonlocal $C^1$-perturbations of a time-periodic parabolic equations and give typical convergence to periodic solutions whose minimal periods are uniformly bounded.