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We consider a dissipative standard map--like system, which is governed by two parameters measuring the strength of the dissipation and of the perturbation. In order to investigate the dynamics, we follow a numerical and an analytical approach. The numerical study relies on the frequency analysis and on the computation of the differential fast Lyapunov indicators. The analytical approach is based on the computation of a suitable normal form for dissipative systems, which allows us to derive an analytic expression of the frequency. \ud    We explore different kinds of attractors (invariant curves, periodic orbits, strange attractors) and their relation with the choice of the perturbing function and of the main frequency of motion (i.e., the frequency of the invariant trajectory of the unperturbed system). In this context we also investigate the occurrence of periodic attractors by looking at the relationship between their periods and the parameters ruling the mapping. Particular attention is devoted to the investigation of the weakly chaotic regime and its transition to the conservative case

Dissipative and weakly--dissipative regimes in nearly--integrable mappings