We study the cosmology of axion-scalar pairs, coupled by a hyperbolic field-space metric and with a string-motivated rational scalar potential. Borrowing tools from the theory of dynamical systems, we are able to classify all late-time trajectories and extract physical properties of the asymptotic solutions. These results suggest a Dynamical Distance Conjecture: along the physical (possibly non-geodesic) trajectories, towers of states become exponentially light as a function of the traversed field-space distance. We further rule out possible counterexamples with wildly oscillating solutions. The considered axion-scalar systems are realized in F-theory compactifications, where the axion-scalar pair is a complex-structure modulus and four-form fluxes induce the asymptotic potentials. We also provide a complete Hodge-theoretic classification of all one-modulus asymptotic potentials of this type.
