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PreprintCC BY 4.0arXiv

Asymptotic Analysis of the Biased van der Pol Equation

  • Alain J. Brizard
  • Desmond L. Carrico
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Abstract

We perform an asymptotic analysis of the biased van der Pol equation ẍ − ν (1 − x2) ẋ + x = a, where the dissipation parameter is large (ν≫ 1) and the bias parameter a is a real number between −1 and $1$, corresponding to large-amplitude relaxation oscillations along a stable limit cycle. Using an abridged singular perturbation analysis, the period of relaxation oscillations T(a,ν) is obtained that is accurate within 0.1% of the numerical results for ν≫ 1.

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Cite

BibTeXRIS
BibTeX
@misc{brizard2026asymptotic,
  title = {{Asymptotic Analysis of the Biased van der Pol Equation}},
  author = {Alain J. Brizard and Desmond L. Carrico},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36398},
  eprint = {2610.36398},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36398},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Asymptotic Analysis of the Biased van der Pol Equation
AU  - Alain J. Brizard
AU  - Desmond L. Carrico
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36398
UR  - https://arxiv.org/abs/2610.36398
AB  - We perform an asymptotic analysis of the biased van der Pol equation ẍ − ν (1 − x^2) ẋ + x = a, where the dissipation parameter is large (ν≫ 1) and the bias parameter a is a real number between −1 and $1$, corresponding to large-amplitude relaxation oscillations along a stable limit cycle. Using an abridged singular perturbation analysis, the period of relaxation oscillations T(a,ν) is obtained that is accurate within 0.1% of the numerical results for ν≫ 1.
ER  -