Abstract
Slow-fast systems on the two-torus are studied. As it was shown before, canard cycles are generic in such systems, which is in drastic contrast with the planar case. It is known that if the rotation number of the Poincaré map is an integer and the slow curve is connected, the number of canard limit cycles is bounded from above by the number of fold points of the slow curve. In the present paper, it is proved that there are no such geometric constraints for non-integer rotation numbers: it is possible to construct a generic system with “as simple as possible” slow curve and arbitrary many limit cycles.
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Notes
The authors are grateful to Victor Kleptsyn for these arguments.
References
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Acknowledgments
The authors are grateful to John Guckenheimer, Victor Kleptsyn, Alexey Klimenko, and Alexey Okunev for fruitful discussions and valuable comments. The authors are especially grateful to Yulij Ilyashenko for his interest in the work and the suggestions on the text of the paper that drastically simplified the proof.
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The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2016–2017 (grant No 16-05-0066) and supported within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program.
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Schurov, I., Solodovnikov, N. Duck Factory on the Two-Torus: Multiple Canard Cycles Without Geometric Constraints. J Dyn Control Syst 23, 481–498 (2017). https://doi.org/10.1007/s10883-016-9335-6
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DOI: https://doi.org/10.1007/s10883-016-9335-6