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Lyapunov spectrum of random neural networks

  • David G. Clark
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Abstract

The Lyapunov spectrum of a nonlinear recurrent neural network with random asymmetric couplings is calculated in the limit N→∞. The calculation is based on a finite-N identity that expresses the cumulative distribution of Lyapunov exponents as a response function of the tangent-space dynamics, with the tangent trajectory selected by a minimum-norm condition rather than by an initial condition. A cavity method then determines this response function at large N through a self-consistent single-site problem. This result establishes that the chaos in this network is extensive and gives access to diffeomorphism-invariant properties of the dynamics. This work was done in collaboration with the AI models GPT-6 Astra and Claude Opus 5.5.

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Cite

BibTeXRIS
BibTeX
@misc{clark2026lyapunov,
  title = {{Lyapunov spectrum of random neural networks}},
  author = {David G. Clark},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12426},
  eprint = {2610.12426},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12426},
}
RIS
TY  - UNPB
TI  - Lyapunov spectrum of random neural networks
AU  - David G. Clark
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12426
UR  - https://arxiv.org/abs/2610.12426
AB  - The Lyapunov spectrum of a nonlinear recurrent neural network with random asymmetric couplings is calculated in the limit N→∞. The calculation is based on a finite-N identity that expresses the cumulative distribution of Lyapunov exponents as a response function of the tangent-space dynamics, with the tangent trajectory selected by a minimum-norm condition rather than by an initial condition. A cavity method then determines this response function at large N through a self-consistent single-site problem. This result establishes that the chaos in this network is extensive and gives access to diffeomorphism-invariant properties of the dynamics. This work was done in collaboration with the AI models GPT-6 Astra and Claude Opus 5.5.
ER  -