Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Uniqueness of relaxation oscillations: A classical approach

Authors: S. P. Hastings and J. B. McLeod
Journal: Quart. Appl. Math. 73 (2015), 201-217
MSC (2010): Primary 34C26; Secondary 37C27
DOI: https://doi.org/10.1090/S0033-569X-2015-01379-6
Published electronically: March 16, 2015
MathSciNet review: 3357492
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Abstract | References | Similar Articles | Additional Information

Abstract: In a recent paper (Kosiuk and Szmolyan (2011)), the authors discuss a relaxation oscillator which apparently has not been put in standard Lienard form. They use geometric perturbation theory to analyze this model. Their main result is the existence and uniqueness of a periodic solution for small values of two parameters, $ \delta $ and $ \varepsilon ,$ and the behavior of this solution as $ \delta $ and $ \varepsilon $ tend to zero. We show how standard ode methods can be used to give shorter and more direct proofs of these results. Along the way we give a new proof of a more general result.

References [Enhancements On Off] (What's this?)

  • [1] Jack K. Hale, Ordinary differential equations, Pure and Applied Mathematics, Volume XXI, Wiley-Interscience [John Wiley & Sons], New York, 1969. MR 0419901 (54 #7918)
  • [2] Ilona Kosiuk and Peter Szmolyan, Scaling in singular perturbation problems: blowing up a relaxation oscillator, SIAM J. Appl. Dyn. Syst. 10 (2011), no. 4, 1307-1343. MR 2854590 (2012i:34074), https://doi.org/10.1137/100814470
  • [3] M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations 174 (2001), no. 2, 312-368. MR 1846739 (2002g:34122), https://doi.org/10.1006/jdeq.2000.3929
  • [4] M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points--fold and canard points in two dimensions, SIAM J. Math. Anal. 33 (2001), no. 2, 286-314 (electronic). MR 1857972 (2002g:34117), https://doi.org/10.1137/S0036141099360919
  • [5] E. F. Mishchenko and N. Kh. Rozov, Differential equations with small parameters and relaxation oscillations, Mathematical Concepts and Methods in Science and Engineering, vol. 13, Plenum Press, New York, 1980. Translated from the Russian by F. M. C. Goodspeed. MR 750298 (85j:34001)
  • [6] Lawrence Perko, Differential equations and dynamical systems, 3rd ed., Texts in Applied Mathematics, vol. 7, Springer-Verlag, New York, 2001. MR 1801796 (2001k:34001)
  • [7] L. Segel and A. Goldbetter, Scaling in biochemical kinetics: Dissection of a relaxation oscillator, J. Math. Biol. 32 (1994), 147-160.

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Additional Information

S. P. Hastings
Affiliation: Department of Mathematics, University of Pittsburgh
Email: sph@pitt.edu

J. B. McLeod
Affiliation: Mathematical Institute, Oxford University

DOI: https://doi.org/10.1090/S0033-569X-2015-01379-6
Received by editor(s): January 12, 2013
Published electronically: March 16, 2015
Additional Notes: The second author is deceased (August 20, 2014).
Article copyright: © Copyright 2015 Brown University

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