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arxiv: 2601.04702 · v2 · submitted 2026-01-08 · ❄️ cond-mat.dis-nn

Chaos in high-dimensional dynamical systems with tunable non-reciprocity

Samantha Fournier , Pierfrancesco Urbani This is my paper

Pith reviewed 2026-05-16 16:48 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords non-reciprocal interactionschaotic attractormaximal Lyapunov exponenthigh-dimensional dynamicssymmetric interactionsgradient descentdisordered systems
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The pith

Any non-reciprocity in interactions drives high-dimensional dynamics onto a chaotic attractor.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies high-dimensional systems of interacting variables whose couplings can be made partially symmetric by tuning a single parameter alpha. Fully symmetric couplings produce gradient descent on a rough landscape and therefore slow, aging dynamics. As soon as alpha allows any directed, non-reciprocal component, the long-time motion settles on a chaotic attractor whose maximal Lyapunov exponent is positive. The exponent varies non-monotonically with alpha, reaching a maximum at intermediate asymmetry. The result shows that even weak conservative forces derived from a disordered energy landscape can be overpowered by small amounts of non-reciprocity and produce sustained chaos.

Core claim

For any value of alpha that introduces non-reciprocal interactions, the dynamics of the high-dimensional system converges to a chaotic attractor characterized by a positive maximal Lyapunov exponent; the exponent itself is a non-monotonic function of alpha. This holds when the interactions are drawn from uncorrelated random distributions in the large-system limit.

What carries the argument

The scalar alpha that continuously interpolates the interaction matrix from fully symmetric (gradient) to fully asymmetric (non-gradient), with the maximal Lyapunov exponent serving as the diagnostic of chaos.

If this is right

  • Chaos appears for arbitrarily small but nonzero non-reciprocity.
  • The maximal Lyapunov exponent peaks at an intermediate value of alpha rather than increasing monotonically with asymmetry.
  • Adding a conservative gradient term from a rough landscape can increase the chaoticity of the motion.
  • The fully symmetric limit replaces chaos with slow relaxation and aging.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Real systems with imperfect reciprocity, such as neural or ecological networks, may generically sit in the chaotic regime.
  • Small engineered asymmetries could be used to enhance exploration in optimization or sampling algorithms.
  • The same transition may occur in finite-dimensional or correlated-interaction versions of the model.

Load-bearing premise

The interactions are drawn independently from random distributions in the infinite-dimensional limit.

What would settle it

Numerically integrate the equations for a large but finite system at very small but nonzero alpha and check whether the maximal Lyapunov exponent remains strictly positive.

Figures

Figures reproduced from arXiv: 2601.04702 by Pierfrancesco Urbani, Samantha Fournier.

Figure 1
Figure 1. Figure 1: FIG. 1: The MLE and the force [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (Left) The correlation function [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (Left) Comparison between numerical simulations on increasing system size (dashed colored lines) and [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Scaling form of the total force driving the dynamical systems Φ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

High-dimensional dynamical systems of interacting degrees of freedom are ubiquitous in the study of complex systems. When the directed interactions are totally uncorrelated, sufficiently strong and non-linear, many of these systems exhibit a chaotic attractor characterized by a positive maximal Lyapunov exponent (MLE). On the contrary, when the interactions are completely symmetric, the dynamics takes the form of a gradient descent on a carefully defined cost function, and it exhibits slow dynamics and aging. In this work, we consider the intermediate case in which the interactions are partially symmetric, with a parameter {\alpha} tuning the degree of non-reciprocity. We show that for any value of {\alpha} for which the corresponding system has non-reciprocal interactions, the dynamics lands on a chaotic attractor. Correspondingly, the MLE is a non-monotonous function of the degree of non-reciprocity. This implies that conservative forcing deriving from the gradient field of a rough energy landscape can make the system more chaotic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies high-dimensional dynamical systems with tunable non-reciprocity in the interactions, parameterized by α. It claims that whenever the interactions are non-reciprocal (any α > 0), the long-time dynamics converge to a chaotic attractor with strictly positive maximal Lyapunov exponent (MLE), while the MLE itself is a non-monotonic function of α. This is contrasted with the fully reciprocal (α = 0) limit, which reduces to gradient descent on a rough landscape with aging, and the fully asymmetric limit, which is chaotic.

Significance. If the central claim holds, the work shows that arbitrarily weak non-reciprocity is sufficient to induce chaos even when a conservative gradient component is present, and that an optimal intermediate asymmetry maximizes the MLE. This provides a concrete mechanism linking partial symmetry breaking to chaotic attractors in high-dimensional systems and could inform models of neural dynamics, ecological networks, or spin glasses with asymmetric couplings.

major comments (2)
  1. [Central claim / Results section] The universality statement that the MLE is positive for any α > 0 rests on the N → ∞ limit with uncorrelated random interactions; the manuscript must supply the explicit mean-field or random-matrix calculation of the Lyapunov spectrum (or the associated dynamical mean-field equations) that demonstrates the absence of a stable fixed-point or periodic regime at small but finite α.
  2. [MLE(α) results] The non-monotonic dependence of the MLE on α is asserted without reported error bars, finite-N scaling, or comparison to the fully asymmetric (α = 1) baseline; numerical or analytic evidence that the non-monotonicity survives the thermodynamic limit and is not an artifact of finite-size fluctuations is required.
minor comments (2)
  1. [Model] The precise definition of the interaction matrix (how the symmetric and antisymmetric parts are sampled and normalized) should be stated as an explicit equation early in the model section.
  2. [Methods] Clarify whether the reported MLE is obtained from direct integration of the tangent dynamics or from a replica-symmetric or cavity-method calculation of the Jacobian spectrum.

Circularity Check

0 steps flagged

No significant circularity detected; chaos claim follows from model dynamics without reduction to inputs

full rationale

The paper presents the result that non-reciprocal interactions (any α > 0) lead to a chaotic attractor with positive MLE as a direct consequence of the high-dimensional limit with uncorrelated random couplings, without any quoted equations showing the MLE or attractor property being fitted to itself, defined circularly, or forced by a self-citation chain. The non-monotonic MLE(α) is reported as an outcome of the dynamics rather than a renaming or ansatz smuggling. No load-bearing steps reduce by construction to the inputs; the derivation remains self-contained under the stated assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on standard assumptions of high-dimensional random interactions with a tunable symmetry parameter; no new particles or forces are introduced.

free parameters (1)
  • alpha
    Tunable control parameter for the degree of non-reciprocity; not fitted to data but chosen to interpolate between limits.
axioms (2)
  • domain assumption Interactions are drawn from uncorrelated random distributions
    Stated in the setup for both fully symmetric and fully asymmetric cases.
  • domain assumption High-dimensional limit applies
    Required for the claimed universality of the chaotic attractor.

pith-pipeline@v0.9.0 · 5468 in / 1257 out tokens · 63786 ms · 2026-05-16T16:48:07.458782+00:00 · methodology

discussion (0)

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Reference graph

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