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We perform a detailed study of the phase-space of the field equations of anEinstein-Gauss-Bonnet scalar field cosmology for a spatially flatFriedmann--Lema\^{\i}tre--Robertson--Walker spacetime. For the scalar fieldpotential, we consider the exponential function. In contrast, for the couplingfunction of the scalar field with the Gauss-Bonnet term, we assume two cases,the exponential function and the power-law function. We write the fieldequations in dimensionless variables and study the equilibrium points usingPoincare variables. For the exponential coupling function, the asymptoticsolutions describe de Sitter universes or spacetimes where the Gauss-Bonnetterm dominates. We recovered previous results but found new asymptoticsolutions not previously studied. For the power-law coupling function,equilibrium points which describe the scaling solution appear. Finally, thepower-law coupling provides a rich cosmological phenomenology.

Phase-space analysis of an Einstein-Gauss-Bonnet scalar field cosmology