We consider transitory canard cycles that consist of a generic breaking mechanism, i.e. a Hopf or a jump breaking mechanism, in combination with a fast–fast passage through a jump point. Such cycle separates two types of canard cycles with a different shape. We obtain upper bounds on the number of periodic orbits that can appear near the canard cycle, and this under very general conditions.
On considère des cycles canard transitoires comportant un mécanisme de cassure générique, de type Hopf ou bien de saut, en combinaison avec un passage de type rapide–rapide par un point de saut. De tels cycles séparent deux types de cycles canard de formes différentes. On obtient des bornes supérieures sur le nombre dʼorbites périodiques qui peuvent apparaître près du cycle canard, sous certaines conditions très générales.
Accepted:
Published online:
Peter De Maesschalck 1; Freddy Dumortier 1; Robert Roussarie 2
@article{CRMATH_2014__352_1_27_0, author = {Peter De Maesschalck and Freddy Dumortier and Robert Roussarie}, title = {Canard-cycle transition at a fast{\textendash}fast passage through a jump point}, journal = {Comptes Rendus. Math\'ematique}, pages = {27--30}, publisher = {Elsevier}, volume = {352}, number = {1}, year = {2014}, doi = {10.1016/j.crma.2013.09.002}, language = {en}, }
TY - JOUR AU - Peter De Maesschalck AU - Freddy Dumortier AU - Robert Roussarie TI - Canard-cycle transition at a fast–fast passage through a jump point JO - Comptes Rendus. Mathématique PY - 2014 SP - 27 EP - 30 VL - 352 IS - 1 PB - Elsevier DO - 10.1016/j.crma.2013.09.002 LA - en ID - CRMATH_2014__352_1_27_0 ER -
Peter De Maesschalck; Freddy Dumortier; Robert Roussarie. Canard-cycle transition at a fast–fast passage through a jump point. Comptes Rendus. Mathématique, Volume 352 (2014) no. 1, pp. 27-30. doi : 10.1016/j.crma.2013.09.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.09.002/
[1] P. De Maesschalck, F. Dumortier, R. Roussarie, Canard cycle transition at a slow–fast passage through a jump point, submitted for publication.
[2] P. De Maesschalck, F. Dumortier, R. Roussarie, Canard cycles from birth to transition, in preparation.
[3] Cyclicity of common slow–fast cycles, Indag. Math. (N. S.), Volume 22 (2011) no. 3–4, pp. 165-206
[4] Slow divergence integral and balanced canard solutions, Qual. Theory Dyn. Syst., Volume 10 (2011) no. 1, pp. 65-85
[5] Multiple canard cycles in generalized Liénard equations, J. Differential Equations, Volume 174 (2001) no. 1, pp. 1-29
[6] Birth of canard cycles, Discrete Contin. Dyn. Syst. Ser. S, Volume 2 (2009) no. 4, pp. 723-781
[7] Relaxation oscillation and canard explosion, J. Differential Equations, Volume 174 (2001) no. 2, pp. 312-368
Cited by Sources:
Comments - Policy