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The dud canard: Existence of strong canard cycles in [formula omitted].
- Source :
-
Journal of Differential Equations . Dec2023, Vol. 375, p706-749. 44p. - Publication Year :
- 2023
-
Abstract
- In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in R 3 through the folded saddle-node of type II and the singular Hopf bifurcation. In particular, we prove – in the analytic case only – that for all 0 < ϵ ≪ 1 there is a family of periodic orbits, born in the (singular) Hopf bifurcation and extending to O (1) cycles that follow the strong canard of the folded saddle-node. Our results can be seen as an extension of the canard explosion in R 2 , but in contrast to the planar case, the family of periodic orbits in R 3 is not explosive. For this reason, we have chosen to call the phenomena in R 3 , the "dud canard". The main difficulty of the proof lies in connecting the Hopf cycles with the canard cycles, since these are described in different scalings. As in R 2 , we use blowup to overcome this, but we also have to compensate for the lack of uniformity near the Hopf bifurcation, due to its singular nature; it is a zero-Hopf bifurcation in the limit ϵ = 0. In the present paper, we do so by imposing analyticity of the vector-field. This allows us to prove existence of an invariant slow manifold, that is not normally hyperbolic. [ABSTRACT FROM AUTHOR]
- Subjects :
- *HOPF bifurcations
*INVARIANT manifolds
*ORBITS (Astronomy)
*LIMIT cycles
Subjects
Details
- Language :
- English
- ISSN :
- 00220396
- Volume :
- 375
- Database :
- Academic Search Index
- Journal :
- Journal of Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 172868945
- Full Text :
- https://doi.org/10.1016/j.jde.2023.09.008