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

TL;DR

This paper constructs an explicit quasipotential for a nonreciprocal stochastic system with two degrees of freedom, revealing complex aperiodic cycles and rich nonequilibrium landscape features, advancing understanding of rare event geometry.

Contribution

It provides the first analytical quasipotential for a nonreciprocal system with two degrees of freedom, capturing complex dynamics and landscape structures.

Findings

Recurrent aperiodic Escher cycles observed

Analytical quasipotential constructed to first order in nonreciprocality

Landscape features flat regions, plateaus, and non-differentiable lines

Abstract

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.