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We examine the cosmological dynamics of $f(R+\mu G)$ gravity models, where $R$ is a Ricci scalar, $G$ is a Gauss-Bonnet term, $\mu$ is a constant, in a four-dimensional spatially flat FLRW metric. We generalize the effective potential method to get de Sitter solutions. We also introduce the dimensionless variables and get dynamical system. We analyze stationary solutions in terms of the introduced dimensionless variables for monomial functions $f(R+\mu G)$.