In this article we perform dynamical analysis of a broad class of barotropic fluids in the spatially curved Friedmann-Robertson-Walker (FRW) spacetime background without considering the cosmological constant. The first part of our study concerns the dynamics of a fluid with an unspecified barotropic equation of state (EoS) having as the only assumption the non-negativity of the fluid's energy density. After defining a new set of dimensionless variables and a new evolution parameter, we

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... the function Γ that encodes the EoS. In this general setup several features of the system are identified: critical points, invariant subsets and the characteristics of the function Γ, along with their cosmological interpretations. The second part of our work provides two examples with specific Γ functions. In the first example we provide a Γ function and then we exhibit how it can be trimmed down to a specific class of EoS through physical arguments, while in the second example we discuss the quadratic EoS studied in Phys.Rev. D 74, 023523 (2006) by comparing our approach with their analysis.