Fate of Arnold tongues and canards in Mobius systems with noise
Abstract
We consider a class of noisy dynamical systems on the two-dimensional torus whose phase evolution is governed by Mobius transformations. In the deterministic limit, these systems favor phase-locking that is described by the band-gap structure of an associated Schrodinger equation with a periodic potential. We use the Wentzel-Kramers-Brillouin method to characterize this structure in the slow-fast limit. Then we show that large-deviation theory describes fluctuations of the finite-time rotation number, whereas weak-noise escape actions determine the exponential scale of rare phase slips. These estimates provide quantitative criteria for metastable phase-locking and melting of Arnold tongues over a prescribed observation time. We derive sufficient conditions for stochastic trajectories to follow prescribed canard segments. With controlled entrance errors, exponential amplification of fluctuations along repelling branches leads to an exponentially small sufficient noise scale for canard persistence in the slow-forcing limit.
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BibTeX
@misc{alexandrov2026fate,
title = {{Fate of Arnold tongues and canards in Mobius systems with noise}},
author = {Artem Alexandrov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26354},
eprint = {2610.26354},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26354},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fate of Arnold tongues and canards in Mobius systems with noise
AU - Artem Alexandrov
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26354
UR - https://arxiv.org/abs/2610.26354
AB - We consider a class of noisy dynamical systems on the two-dimensional torus whose phase evolution is governed by Mobius transformations. In the deterministic limit, these systems favor phase-locking that is described by the band-gap structure of an associated Schrodinger equation with a periodic potential. We use the Wentzel-Kramers-Brillouin method to characterize this structure in the slow-fast limit. Then we show that large-deviation theory describes fluctuations of the finite-time rotation number, whereas weak-noise escape actions determine the exponential scale of rare phase slips. These estimates provide quantitative criteria for metastable phase-locking and melting of Arnold tongues over a prescribed observation time. We derive sufficient conditions for stochastic trajectories to follow prescribed canard segments. With controlled entrance errors, exponential amplification of fluctuations along repelling branches leads to an exponentially small sufficient noise scale for canard persistence in the slow-forcing limit.
ER -