Noise-induced transitions between coexisting limit cycles beyond saddle-node bifurcations
Abstract
Nonlinear multistable systems with limit cycles may exhibit noise-induced transitions between oscillatory states, even in parameter regimes beyond the saddle-node bifurcations that determine the existence of these limit cycles. This behavior is associated with the ghost of a saddle-node bifurcation, a well-known feature of nonlinear systems near such bifurcation points. To investigate this phenomenon, we first consider a stochastic minimal model that supports two coexisting limit cycles within a parameter interval bounded by two saddle-node bifurcations. We numerically and theoretically estimate the probability of remaining near each stable attractor as a function of both the noise intensity and the distance to the ghost structure. The model is then extended by introducing spatially localized, non-homogeneous noise with enhanced intensity in a prescribed region of phase space. This modification is intended to emulate the asymmetric structure of the Hindmarsh-Rose model, in which bursting limit cycles approach each other closely in specific regions of phase space, thereby facilitating noise-induced transitions between them.
Versions
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BibTeX
@misc{ortegapiwonka2026noiseinduced,
title = {{Noise-induced transitions between coexisting limit cycles beyond saddle-node bifurcations}},
author = {Ignacio Ortega-Piwonka and Javier Used and Jesús M. Seoane and Miguel A. F. Sanjuán},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23235},
eprint = {2610.23235},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23235},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Noise-induced transitions between coexisting limit cycles beyond saddle-node bifurcations
AU - Ignacio Ortega-Piwonka
AU - Javier Used
AU - Jesús M. Seoane
AU - Miguel A. F. Sanjuán
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23235
UR - https://arxiv.org/abs/2610.23235
AB - Nonlinear multistable systems with limit cycles may exhibit noise-induced transitions between oscillatory states, even in parameter regimes beyond the saddle-node bifurcations that determine the existence of these limit cycles. This behavior is associated with the ghost of a saddle-node bifurcation, a well-known feature of nonlinear systems near such bifurcation points. To investigate this phenomenon, we first consider a stochastic minimal model that supports two coexisting limit cycles within a parameter interval bounded by two saddle-node bifurcations. We numerically and theoretically estimate the probability of remaining near each stable attractor as a function of both the noise intensity and the distance to the ghost structure. The model is then extended by introducing spatially localized, non-homogeneous noise with enhanced intensity in a prescribed region of phase space. This modification is intended to emulate the asymmetric structure of the Hindmarsh-Rose model, in which bursting limit cycles approach each other closely in specific regions of phase space, thereby facilitating noise-induced transitions between them.
ER -