  Diverse equilibrium systems with heterogeneous interactions lie at the edge
of stability. Such marginally stable states are dynamically selected as the
most abundant ones or as those with the largest basins of attraction. On the
other hand, systems with non-reciprocal (or asymmetric) interactions are
inherently out of equilibrium, and exhibit a rich variety of steady states,
including fixed points, limit cycles and chaotic trajectories. How are steady
states dynamically selected away from equilibrium? We address this question in
a simple neural network model, with a tunable level of non-reciprocity. Our
study reveals different types of ordered phases and it shows how
non-equilibrium steady states are selected in each phase. In the spin-glass
region, the system exhibits marginally stable behaviour for reciprocal (or
symmetric) interactions and it smoothly transitions to chaotic dynamics, as the
non-reciprocity (or asymmetry) in the couplings increases. Such region, on the
other hand, shrinks and eventually disappears when couplings become
anti-symmetric. Our results are relevant to advance the knowledge of disordered
systems beyond the paradigm of reciprocal couplings, and to develop an
interface between statistical physics of equilibrium spin-glasses and dynamical
systems theory.
