Abstract
This article deals with relaxation oscillations from a generic balanced canard cycle \(\Gamma\) subject to three breaking parameters of Hopf or jump type. We prove that in a rescaled layer of \(\Gamma\) there bifurcate at most five relaxation oscillations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
De Maesschalk, P., Dumortier, F.: Time analysis and entry-exit relation near planar turning point. J. Differ. Equ. 215, 225–267 (2005)
Dumortier, F., Roussarie, R.: Canard cycles and centre manifolds. Mem. Am. Math. Soc. 121 (577), 1–100 (1996)
Dumortier, F., Roussarie, R.: Multiple canard cycles in generalized Liénard equations. J. Differ. Equ. 174, 1–29 (2001)
Dumortier, F., Roussarie, R.: Canard cycles with two breaking parameters. Discrete Continuous Dyn. Syst. 17 (4), 787–806 (2007)
Dumortier, F., Roussarie, R.: Multi-layer canard cycles and translated power functions. J. Differ. Equ. 244, 1329–1358 (2008)
Khovanskii, A.: Fewnomials. Translated from the Russian by Smilka Zdravkovska. Translations of Mathematical Monographs, vol. 88, viii + 139 pp. American Mathematical Society, Providence (1991)
Mahmoudi, L., Roussarie, R.: Canard cycles of finite codimension with two breaking parameters. Qual. Theory Dyn. Syst. 11, 167–198 (2012)
Panazzolo, D.: Solutions of the equation \(a_{n} + (a_{n-1} +\ldots (a_{2} + (a_{1} + x^{r_{1}})^{r_{2}}\ldots )^{r_{n}} = x\), oral communication (2015)
Acknowledgements
The first author is supported by Ramon y Cajal grant RYC-2011-07730, and also partially by grants MINECO/FEDER MTM2008-03437, MINECO MTM2013-40998-P, and AGAUR 2014SGR-568.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Caubergh, M., Roussarie, R. (2016). Canard Cycles with Three Breaking Mechanisms. In: Toni, B. (eds) Mathematical Sciences with Multidisciplinary Applications. Springer Proceedings in Mathematics & Statistics, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-31323-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-31323-8_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31321-4
Online ISBN: 978-3-319-31323-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)