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

TL;DR

This paper addresses rare-event statistics in a minimal two-dimensional stochastic system with nonreciprocal coupling, exhibiting Escher cycles—noise-driven transitions among four metastable states. It develops a full Freidlin–Wentzell quasipotential framework, first constructing relative quasipotentials in 1D and then assembling the complete 2D quasipotential from competing fluctuation paths (direct and indirect) and their saddles, revealing flat regions, plateaus, and non-differentiable switching lines. A perturbative approach to first order in the nonreciprocal coupling $\gamma$ is validated against numerical calculations, and a reduced-symmetry generalisation demonstrates how basin weights adapt under asymmetric driving. The results provide a transparent analytic example of a nonequilibrium quasipotential in multiple coordinates, offering insight into nonreciprocal nucleation and guiding future analyses of more complex spatially extended systems.

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.

Figures (12)

• Figure 1: The model in Eq. \ref{['eq:model']} and its fixed points for $\gamma / \gamma_{\rm c} = 0.5$. The thin grey lines show the flow field of the deterministic drift $\bm{R}$. The filled markers show the four stable fixed points $\mathrm{A} , \mathrm{B} , \mathrm{C} , \mathrm{D}$, and the hollow markers show the five unstable fixed points $\mathcal{U} = \{ \mathrm{O} , \mathrm{P} , \mathrm{Q} , \mathrm{R} , \mathrm{S} \}$. The smaller grey markers show the equilibrium fixed points for $\gamma = 0$ as a visual guide. The rotating arrow indicates the preferred direction of rotation for positive interaction $\gamma > 0$.
• Figure 2: Projection of the $2d$ system onto the $1d$ reaction coordinate $\varsigma$. Left: Instantons (orange) between attractors $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$, and $\mathrm{D}$ in clockwise direction for small interaction strength $\gamma$. Centre: Reaction coordinate $\varsigma$ obtained by projecting the instantons from the $2d$ system. Note that the reaction coordinate is periodic. Right: Global QP $F(\varsigma)$ visualised as a height variable above the $1d$ reaction coordinate which lies in the $2d$ configuration space $(x,y)$.
• Figure 3: Relative and global QPs in the $1d$ reduction as a function of $\varsigma$ for $\gamma/ \gamma_\mathrm{c} = 0.2$. (a) Construction of the clockwise (+) and counter-clockwise (-) branches of the RQP $F^\pm_X$. Thick lines indicate initial construction between $\mathrm{A}$ and $\mathrm{B}$, thin line shows the periodic extension. (b) Construction of the RQP $F_ \mathrm{A}$ (highlighted by a thick light red line with shaded region) as the minimum of both branches. The circle marks a first-order singularity in $F_ \mathrm{A}$. (c) The global QP $F(\varsigma)$ (shown as a thick dark red line with shading), constructed as the minimum over the four RQPs $F_X(\varsigma)$ associated with the stable fixed points $X \in \{ \mathrm{A} , \mathrm{B} , \mathrm{C} , \mathrm{D} \}$. The RQPs $F_X$ for $X = \mathrm{B} , \mathrm{C} , \mathrm{D}$ are obtained from $F_ \mathrm{A}$ via translation in space. The squares mark the first-order singularities in $F$ which are located away from saddles and minima, both of which are marked by the dotted vertical lines.
• Figure 4: Direct vs. indirect stationary paths in the $2d$ system connecting the stable fixed point $\mathrm{A}$ with targets $\bm{z}$. These are the instantons that were computed numerically for $\gamma/\gamma_c=0.1$ and $\bm{z} = (-0.243, 0.2)$ and $(-0.242, 0.2)$. The direct path connects $\mathrm{A}$ and $\bm{z}$ directly, while the indirect path passes through the unstable fixed point $\mathrm{P}$ on the way. For each of the two points, direct and indirect paths both exist as local minima of the action; we only show the global minimum in each case, to illustrate its switch between the branches.
• Figure 5: Construction of RQPs based on the two branches of the RQP for $\gamma = 0.1 \gamma_\mathrm{c}$. The contours are equally spaced between $0.05$ and $0.6$, and include the saddle contour $\frac{1}{4} - 0.1 \gamma_\mathrm{c} \approx 0.216$. Left: Direct branch of the RQP. The red shaded region highlights parts of the direct branch that break monotonicity in $x$-direction. Centre: Indirect branch of the RQP. Right: The RQP $F_ \mathrm{A}$, showing the regions where it derives from each branch. The black line marks the boundary where the instanton path switches from direct to indirect. The thick part of this line highlights the first-order part region where indirect and direct paths are distinct, see also Fig. \ref{['fig:path_dissimilarity']}.