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Herein, we consider the dynamics of $C^1$-perturbed almost periodic systemswith strong cone conditions in the sketch of skew-product semiflows. With somesuitable dissipative assumptions, we investigate the asymptotic behaviors ofany orbit of such perturbed systems and we obtain that any omega-limit setgenerated by the skew-product semiflow of the perturbed system contains at mosttwo minimal sets, and any minimal set is an almost automorphic extension of itsbase flow. These results reveal that the structures of omega-limit sets andminimal sets for unperturbed systems can carry out to the systems with small$C^1$-perturbations. Furthermore, we apply these theoretic results to sometypical models, such as scalar parabolic equations with separated boundaryconditions and competitive-cooperative tridiagonal systems.

Dynamics of $C^1$-perturbed non-autonomous systems with strong cone conditions and its applications