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PreprintarXiv licencearXiv

Exponential Stability of Slow--Fast Stochastic Systems with a Small Delay

  • Yudong Wang
  • Kele Sun
  • Shijie Zhou
  • Xuerong Mao
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Abstract

In this paper, we investigate the exponential stability of coupled slow--fast stochastic systems with a small delay. Under an averaged Lyapunov condition, we establish practical pth-moment exponential stability for 0<p<2 using a cutoff-function approach to handle singular Lyapunov derivatives near the origin. Exponential stability in the pth moment and almost surely is further obtained through two distinct mechanisms: homogeneous dissipation of the fast dynamics or vanishing sensitivity of the slow coefficients to the fast state near the origin. We extend these results to locally Lipschitz systems, establishing exponential stability with arbitrarily high probability for bounded sets of initial data, thereby broadening the applicability of stability analysis for stochastic systems with delayed feedback. Numerical examples illustrate our results.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026exponential,
  title = {{Exponential Stability of Slow--Fast Stochastic Systems with a Small Delay}},
  author = {Yudong Wang and Kele Sun and Shijie Zhou and Xuerong Mao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19008},
  eprint = {2610.19008},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19008},
}
RIS
TY  - UNPB
TI  - Exponential Stability of Slow--Fast Stochastic Systems with a Small Delay
AU  - Yudong Wang
AU  - Kele Sun
AU  - Shijie Zhou
AU  - Xuerong Mao
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.19008
UR  - https://arxiv.org/abs/2610.19008
AB  - In this paper, we investigate the exponential stability of coupled slow--fast stochastic systems with a small delay. Under an averaged Lyapunov condition, we establish practical pth-moment exponential stability for 0<p<2 using a cutoff-function approach to handle singular Lyapunov derivatives near the origin. Exponential stability in the pth moment and almost surely is further obtained through two distinct mechanisms: homogeneous dissipation of the fast dynamics or vanishing sensitivity of the slow coefficients to the fast state near the origin. We extend these results to locally Lipschitz systems, establishing exponential stability with arbitrarily high probability for bounded sets of initial data, thereby broadening the applicability of stability analysis for stochastic systems with delayed feedback. Numerical examples illustrate our results.
ER  -