Extending the Dynamical Systems Toolkit: Coupled Fields in Multiscalar Dark Energy

Abstract

We study the dynamics of a two-field scalar model consisting of an axion-saxion pair with both kinetic and potential couplings, as motivated by string theory compactifications. We extend the dynamical systems (DS) toolkit by introducing a new set of variables that not only close the system and enable a systematic stability analysis, but also disentangle the role of the kinetic coupling. Within this framework we derive a compact, general expression for the non-geodesicity (turning-rate) parameter evaluated at fixed points, valid for arbitrary couplings. This provides a transparent way of diagnosing non-geodesic dynamics, with direct applications to both dark energy and multifield inflation. We first consider exponential coupling functions to establish analytic control and facilitate comparison with previous literature. In this case, we uncover a pair of genuinely non-geodesic fixed points, which act as attractors within a submanifold of the full system. In contrast, when the axion shift symmetry remains unbroken, our analysis shows that the apparent non-geodesic fixed point reported previously does not persist once the full dynamics are taken into account. Finally, we illustrate how our approach naturally extends to more realistic string-inspired models, such as power-law axion potentials combined with exponential saxion couplings, and present an explicit supergravity realisation.