Asymptotic dynamical analysis of $f(R,T^{\phi}) = R+\alpha T^{\phi} + \beta (T^{\phi})^2/2$ cosmology

In this work we investigate the asymptotic cosmological dynamics of a modified gravity model based on the $f(R,T^\phi)$ theory, where $R$ denotes the Ricci scalar and $T^\phi$ is the trace of the stress-energy tensor of a scalar field. Despite the extensive study of $f(R,T)$ gravity, the asymptotic implications of quadratic trace couplings in scalar field cosmology remain largely unexplored. We focus on a specific form given by $f(R,T^\phi) = R + \alpha T^\phi + \beta (T^\phi)^2/2$, in which the parameters $\alpha$ and $\beta$ control the strength of non-minimal couplings between geometry and matter. We derive the set of cosmological equations for a spatially flat, homogeneous and isotropic universe and construct the autonomous system of first-order differential equations using a compact set of dimensionless variables. This formulation provides a foundation for the qualitative analysis of the asymptotic behavior. We identify and classify all critical points and analyze their stability properties. Finally, the energy conditions and the presence of dynamical instabilities are examined. We study the general scenario $\alpha \neq 0$ and $\beta \neq 0$, along with the subcases $\alpha = 0$ and $\beta = 0$, in order to compare with minimally coupled quintessence $\alpha = \beta = 0$. We find that the quadratic term in $T^\phi$ admits late-time accelerated de Sitter-like critical solutions at the background level. However, several accelerated points lie in a degenerate scalar sector with $Q_s=0$, where the standard linear perturbation criteria are inconclusive, while the quasi-de Sitter point with $Q_s>0$ is of saddle type. Therefore, establishing full perturbative viability requires going beyond the linear analysis in the degenerate sector.