Nonreciprocal dynamics with weak noise: aperiodic "Escher cycles" and their quasipotential landscape

We present an explicit construction of the Freidlin-Wentzell quasipotential of a stochastic system with two degrees of freedom and nonreciprocal interactions. This model undergoes noise-induced transitions between four metastable attractors, forming recurrent but aperiodic ``Escher cycles,'' similar to the cyclic nucleation dynamics observed in the nonreciprocal Ising model. We calculate the quasipotential analytically to first order in nonreciprocality. We characterise it along a one-dimensional reaction coordinate that connects the attractors, and we also obtain the full two-dimensional landscape, at leading order in perturbation theory. The resulting landscapes feature flat regions and extended plateaus, together with non-differentiable switching lines. These singular structures arise from two geometric mechanisms: the handover of dominance between competing transition paths, and the competition between basins of attraction. The system provides a rare case where the geometry of nonequilibrium rare events can be fully resolved, and a simple analytically tractable example of a quasipotential in more than one coordinate that captures a rich set of nonequilibrium features.