Publications

Papers & preprints.

Work on random dynamical systems, stochastic fluid dynamics, and the dynamics of learning.

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Published articles

2025

Published

The conditioned Lyapunov spectrum for random dynamical systems

Matheus M. Castro, Dennis Chemnitz, Hugo Chu, Maximilian Engel, Jeroen S. W. Lamb, Martin Rasmussen

Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 61(3), 1845–1877

PDF DOI arXiv

We establish a full Lyapunov spectrum for random dynamical systems with absorption. Embedding the process conditioned on indefinite survival into the random dynamical systems framework gives convergence of finite-time exponents in conditioned probability.

BibTeX citation

@article{castro2025conditioned,
  author = {Castro, Matheus M. and Chemnitz, Dennis and Chu, Hugo and Engel, Maximilian and Lamb, Jeroen S. W. and Rasmussen, Martin},
  title = {The conditioned {Lyapunov} spectrum for random dynamical systems},
  journal = {Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques},
  year = {2025},
  volume = {61},
  number = {3},
  pages = {1845--1877},
  doi = {10.1214/24-AIHP1466},
  eprint = {2204.04129},
  archivePrefix = {arXiv}
}

2025

Published

Characterizing Dynamical Stability of Stochastic Gradient Descent in Overparameterized Learning

Dennis Chemnitz, Maximilian Engel

Journal of Machine Learning Research 26(134), 1–46

PDF Journal arXiv

We characterize which interpolation minima are dynamically stable under gradient descent and stochastic gradient descent. A characteristic Lyapunov exponent determines whether SGD can accumulate at a given minimum, connecting local stability to the selection of solutions.

BibTeX citation

@article{chemnitz2025sgdstability,
  author = {Chemnitz, Dennis and Engel, Maximilian},
  title = {Characterizing Dynamical Stability of Stochastic Gradient Descent in Overparameterized Learning},
  journal = {Journal of Machine Learning Research},
  year = {2025},
  volume = {26},
  number = {134},
  pages = {1--46},
  url = {https://jmlr.org/papers/v26/24-1547.html}
}

2023

Published

Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise

Dennis Chemnitz, Maximilian Engel

Communications in Mathematical Physics 402(2), 1807–1843

PDF DOI arXiv

We prove that sufficiently strong shear produces a positive Lyapunov exponent in the Hopf normal form with additive noise. The proof uses a scaling limit and continuity of Lyapunov exponents to establish shear-induced chaos.

BibTeX citation

@article{chemnitz2023hopf,
  author = {Chemnitz, Dennis and Engel, Maximilian},
  title = {Positive {Lyapunov} Exponent in the {Hopf} Normal Form with Additive Noise},
  journal = {Communications in Mathematical Physics},
  year = {2023},
  volume = {402},
  number = {2},
  pages = {1807--1843},
  doi = {10.1007/s00220-023-04764-z},
  eprint = {2212.06547},
  archivePrefix = {arXiv}
}

Preprints

2026

Preprint

Stochastic Flows with Strong Shear - Part I: Strong Completeness and Set Attractors

Dennis Chemnitz, Maximilian Engel, Michael Scheutzow

arXiv:2610.06321

PDF arXiv

We study how strong, radius-dependent shear affects two-dimensional stochastic flows. The paper gives sufficient and necessary conditions for strong completeness and the existence of set attractors, and shows how sufficiently fast growth of the shear can cause exceptional initial conditions to escape to infinity in finite time.

BibTeX citation

@misc{chemnitz2026strongshear,
  author = {Chemnitz, Dennis and Engel, Maximilian and Scheutzow, Michael},
  title = {Stochastic Flows with Strong Shear - Part I: Strong Completeness and Set Attractors},
  year = {2026},
  eprint = {2610.06321},
  archivePrefix = {arXiv},
  primaryClass = {math.PR},
  url = {https://arxiv.org/abs/2610.06321}
}

2025

Preprint

Mixing at the Batchelor Scale for White-In-Time Flows

Robin Chemnitz, Dennis Chemnitz

arXiv:2512.04297 · Revised June 2026

PDF arXiv

We study passive scalar mixing in smooth, white-in-time flows. Bounds on the almost-sure exponential dissipation rate establish Batchelor-scale behavior in explicit two- and three-dimensional models, including estimates on mixing without diffusion.

BibTeX citation

@misc{chemnitz2025batchelor,
  author = {Chemnitz, Robin and Chemnitz, Dennis},
  title = {Mixing at the {Batchelor} Scale for White-In-Time Flows},
  year = {2025},
  eprint = {2512.04297},
  archivePrefix = {arXiv},
  primaryClass = {math.PR},
  doi = {10.48550/arXiv.2512.04297},
  url = {https://arxiv.org/abs/2512.04297}
}

2025

Preprint

A Dynamical Systems Perspective on the Analysis of Neural Networks

Dennis Chemnitz, Maximilian Engel, Christian Kuehn, Sara-Viola Kuntz

arXiv:2507.05164 · Book chapter preprint · Revised June 2026

PDF arXiv

An expository dynamical systems perspective on neural networks: information propagation, training and stability, and mean-field limits. The chapter connects neural ODEs, gradient descent, and stochastic learning to a common mathematical framework.

BibTeX citation

@misc{chemnitz2025neuralnetworks,
  author = {Chemnitz, Dennis and Engel, Maximilian and Kuehn, Christian and Kuntz, Sara-Viola},
  title = {A Dynamical Systems Perspective on the Analysis of Neural Networks},
  year = {2025},
  eprint = {2507.05164},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  doi = {10.48550/arXiv.2507.05164},
  url = {https://arxiv.org/abs/2507.05164}
}

Dissertation

2025

Dissertation

Stability of Random Dynamical Systems in Noisy Oscillators with Shear and in Stochastic Gradient Descent

Dennis Chemnitz

PhD thesis · Freie Universität Berlin

Repository

My dissertation studies stability, shear-induced chaos and blowup, stochastic flows, and synchronization, alongside the dynamical stability of gradient-based learning algorithms.

BibTeX citation

@phdthesis{chemnitz2025thesis,
  author = {Chemnitz, Dennis},
  title = {Stability of Random Dynamical Systems in Noisy Oscillators with Shear and in Stochastic Gradient Descent},
  school = {Freie Universit\"at Berlin},
  year = {2025},
  doi = {10.17169/refubium-47267},
  url = {https://refubium.fu-berlin.de/handle/fub188/47549}
}

Publication details link to the journal, arXiv, or institutional repository record. Preprints are listed separately from published articles.

Published articles

The conditioned Lyapunov spectrum for random dynamical systems

Characterizing Dynamical Stability of Stochastic Gradient Descent in Overparameterized Learning

Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise

Preprints

Stochastic Flows with Strong Shear - Part I: Strong Completeness and Set Attractors

Mixing at the Batchelor Scale for White-In-Time Flows

A Dynamical Systems Perspective on the Analysis of Neural Networks

Dissertation

Stability of Random Dynamical Systems in Noisy Oscillators with Shear and in Stochastic Gradient Descent

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