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.
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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.
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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
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DOI: https://doi.org/10.1007/978-3-319-31323-8_4
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