Sharpened dynamics alternative and its C1-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 celebrated Hirsch's limit-set dichotomy continuous-time semiflows. In this paper, we first present sharpened C 1 -smooth dissipative system { F 0 n } ∈ N (with an attractor A ), which concludes that there is positive integer m such any orbit either manifestly unstable; or asymptotic linearly stable cycle whose minimal period bounded by . Furthermore, show -robustness alternative, is, -perturbed ϵ ( not necessarily monotone), initiated nearby will admit with same The improved generic convergence cycles -system , as well perturbed thus obtained by-products and its -robustness. results are applied nonlocal -perturbations time-periodic parabolic equations give typical periodic solutions periods uniformly bounded.