Sharpened dynamics alternative and its C1-robustness for strongly monotone discrete dynamical systems

Abstract

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 C1-smooth strongly monotone discrete-time dissipative system {F0n}n∈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 C1-robustness of the sharpened dynamics alternative, that is, for any C1-perturbed system {Fϵn}n∈N (Fϵ 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 C1-system {F0n}n∈N, as well as for the perturbed system {Fϵn}n∈N, is thus obtained as by-products of the sharpened dynamics alternative and its C1-robustness. The results are applied to nonlocal C1-perturbations of a time-periodic parabolic equations and give typical convergence to periodic solutions whose minimal periods are uniformly bounded.