We study the effects of a finite kicked environment (bath) composed of N harmonic oscillators on the particle transport in a weakly dissipative quasisymmetric potential system. The small spatial asymmetry is responsible for the appearance of directed particle transport without a net bias, known as the ratchet transport. The whole dynamics is governed by a generalized map where dissipation in the system emerges due to its interaction with the kicked environment. Distinct spectral densities are imposed to the bath oscillators and play an essential role in such models. By changing the functional form of the spectral density, we observe that the transport can be optimized or even suppressed. We show evidences that the transport optimization is related to stability properties of periodic points of the ratchet system and depends on the bath temperature. In a Markovian approach, transport can be increased or suppressed depending on the bath influence.
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