Stability analysis of dilaton-inspired scalar field within the geometrical trinity of gravity

We investigate the dynamics of the dilaton-inspired scalar field, formally rewritten by means of a Brans-Dicke Lagrangian, within the framework of \emph{geometrical trinity of gravity}. In this respect, we perform a stability analysis by adopting a non-flat Friedmann-Robertson-Walker (FRW) metric and considering the well-established exponential potential in three distinct gravitational frameworks: general relativity, teleparallel gravity, and symmetric-teleparallel gravity. By comparing the scalar field behaviors across these theories, we highlight the role of curvature, torsion, and non-metricity in shaping cosmic evolution. Our analysis reveals that, both in general relativity and teleparallel gravity, the dilaton-inspired field can drive the accelerated expansion of the universe, effectively behaving as cosmological constant at late times. In contrast, within the symmetric teleparallel gravity scenario, performing a complete linear stability analysis is prevented by the use of the non-coincident gauge. Nevertheless, the latter paradigm introduces complexity into the autonomous system, resulting in a structurally different analysis. For general relativity and teleparallel scenarios, we remark the regions of attractor solutions and unphysical domains in which we do not expect the viability of our dilaton-inspired Lagrangian. However, within the framework of symmetric-teleparallel gravity, the stability analysis reveals no attractor points for the chosen set of free parameters. In support of these findings, physical conclusions, kinematical studies, and consequences on Friedmann dynamics are thus explored.