Global dynamics in Einstein-Gauss-Bonnet scalar field cosmology with matter

We study the dynamics of the field equations in a four-dimensional isotropic andhomogeneous spatially flat Friedmann-Lemaitre-Robertson-Walker geometry in the context of Einstein-Gauss-Bonnet theory with a matter source and a scalar field coupled to the Gauss-Bonnet scalar. In this theory, the Gauss-Bonnet term contributes to the field equations. The mass of the scalar field depends on the potential function and the Gauss-Bonnet term. For the scalar field potential, we consider the exponential function and the coupling function between the scalar field and the Gauss-Bonnet scalar is considered to be the linear function. Moreover, the scalar field can have a phantom behavior. We consider a set of dimensionless variables and write the field equations into a system or algebraic-differential equations. For the latter, we investigate the equilibrium points and their stability properties. We use compactified variables to perform a global analysis of the asymptotic dynamics. This gravitational theory can explain the Universe's recent and past acceleration phases. Therefore, it can be used as a toy model for studying inflation or as a dark energy candidate.