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For strongly monotone dynamical systems, the dynamics alternative for smoothdiscrete-time systems turns out to be a perfect analogy of the celebratedHirsch's limit-set dichotomy for continuous-time semiflows. In this paper, wefirst present a sharpened dynamics alternative for $C^1$-smooth stronglymonotone discrete-time dissipative system $\{F_0^n\}_{n\in \mathbb{N}}$ (withan attractor $A$), which concludes that there is a positive integer $m$ suchthat any orbit is either manifestly unstable; or asymptotic to a linearlystable 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 $\{F_\epsilon^n\}_{n\in \mathbb{N}}$ ($F_\epsilon$ notnecessarily monotone), any orbit initiated nearby $A$ will admit the sharpeneddynamics alternative with the same $m$. The improved generic convergence tocycles for the $C^1$-system $\{F_0^n\}_{n\in \mathbb{N}}$, as well as for theperturbed system $\{F_\epsilon^n\}_{n\in \mathbb{N}}$, is thus obtained asby-products of the sharpened dynamics alternative and its $C^1$-robustness. Theresults are applied to nonlocal $C^1$-perturbations of a time-periodicparabolic equations and give typical convergence to periodic solutions whoseminimal periods are uniformly bounded.

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