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Pattern formation and dynamic transition for magnetohydrodynamic convection
Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations
| 1. | Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, 3590 Diepenbeek, Belgium |
| 2. | Hasselt University, Campus Diepenbeek, Agoralaan-Gebouw D, B-3590 Diepenbeek, Belgium |
| 3. | Universiteit Hasselt, Campus Diepenbeek, Agoralaan–gebouw D, 3590 Diepenbeek |
References:
| [1] |
P. De Maesschalck and F. Dumortier, Slow-fast Bogdanov-Takens bifurcations, J. Differential Equations, 250 (2011), 1000-1025.
doi: 10.1016/j.jde.2010.07.022. |
| [2] |
F. Dumortier and R. Roussarie, Birth of canard cycles, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 723-781.
doi: 10.3934/dcdss.2009.2.723. |
| [3] |
P. De Maesschalck and F. Dumortier, Singular perturbations and vanishing passage through a turning point, J. Differential Equations, 248 (2010), 2294-2328.
doi: 10.1016/j.jde.2009.11.009. |
| [4] |
P. De Maesschalck and F. Dumortier, Canard cycles in the presence of slow dynamics with singularities, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 265-299.
doi: 10.1017/S0308210506000199. |
| [5] |
F. Dumortier, R. Roussarie, J. Sotomayor, and H. Zoladek, Bifurcations of Planar Vector Fields, Lecture Notes in Mathematics, 1480, Springer-Verlag, Berlin, 1991. |
| [6] |
Freddy Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations, 224 (2006), 296-313.
doi: 10.1016/j.jde.2005.08.011. |
| [7] |
Robert Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst., 17 (2007), 441-448.
doi: 10.3934/dcds.2007.17.441. |
| [8] |
R. Huzak, P. De Maesschalck and F. Dumortier, Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations, J. Differential Equations, 255 (2013), 4012-4051.
doi: 10.1016/j.jde.2013.07.057. |
| [9] |
Peter De Maesschalck and Freddy Dumortier, Detectable canard cycles with singular slow dynamics of any order at the turning point, Discrete Contin. Dyn. Syst., 29 (2011), 109-140.
doi: 10.3934/dcds.2011.29.109. |
| [10] |
P. De Maesschalck and F. Dumortier, Time analysis and entry-exit relation near planar turning points, J. Differential Equations, 215 (2005), 225-267.
doi: 10.1016/j.jde.2005.01.004. |
| [11] |
P. Bonckaert, P. De Maesschalck and F. Dumortier, Well adapted normal linearization in singular perturbation problems, J. Dynam. Differential Equations, 23 (2011), 115-139.
doi: 10.1007/s10884-010-9191-0. |
| [12] |
M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.
doi: 10.1006/jdeq.2000.3929. |
| [13] |
Daniel Panazzolo, Desingularization of nilpotent singularities in families of planar vector fields, Mem. Amer. Math. Soc., 158 (2002).
doi: 10.1090/memo/0753. |
| [14] |
J. Llibre, A survey on the limit cycles of the generalized polynomial Liénard differential equations, Mathematical models in engineering, biology and medicine, AIP Conf. Proc., 1124 (2009), 224-233. |
show all references
References:
| [1] |
P. De Maesschalck and F. Dumortier, Slow-fast Bogdanov-Takens bifurcations, J. Differential Equations, 250 (2011), 1000-1025.
doi: 10.1016/j.jde.2010.07.022. |
| [2] |
F. Dumortier and R. Roussarie, Birth of canard cycles, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 723-781.
doi: 10.3934/dcdss.2009.2.723. |
| [3] |
P. De Maesschalck and F. Dumortier, Singular perturbations and vanishing passage through a turning point, J. Differential Equations, 248 (2010), 2294-2328.
doi: 10.1016/j.jde.2009.11.009. |
| [4] |
P. De Maesschalck and F. Dumortier, Canard cycles in the presence of slow dynamics with singularities, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 265-299.
doi: 10.1017/S0308210506000199. |
| [5] |
F. Dumortier, R. Roussarie, J. Sotomayor, and H. Zoladek, Bifurcations of Planar Vector Fields, Lecture Notes in Mathematics, 1480, Springer-Verlag, Berlin, 1991. |
| [6] |
Freddy Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations, 224 (2006), 296-313.
doi: 10.1016/j.jde.2005.08.011. |
| [7] |
Robert Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst., 17 (2007), 441-448.
doi: 10.3934/dcds.2007.17.441. |
| [8] |
R. Huzak, P. De Maesschalck and F. Dumortier, Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations, J. Differential Equations, 255 (2013), 4012-4051.
doi: 10.1016/j.jde.2013.07.057. |
| [9] |
Peter De Maesschalck and Freddy Dumortier, Detectable canard cycles with singular slow dynamics of any order at the turning point, Discrete Contin. Dyn. Syst., 29 (2011), 109-140.
doi: 10.3934/dcds.2011.29.109. |
| [10] |
P. De Maesschalck and F. Dumortier, Time analysis and entry-exit relation near planar turning points, J. Differential Equations, 215 (2005), 225-267.
doi: 10.1016/j.jde.2005.01.004. |
| [11] |
P. Bonckaert, P. De Maesschalck and F. Dumortier, Well adapted normal linearization in singular perturbation problems, J. Dynam. Differential Equations, 23 (2011), 115-139.
doi: 10.1007/s10884-010-9191-0. |
| [12] |
M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.
doi: 10.1006/jdeq.2000.3929. |
| [13] |
Daniel Panazzolo, Desingularization of nilpotent singularities in families of planar vector fields, Mem. Amer. Math. Soc., 158 (2002).
doi: 10.1090/memo/0753. |
| [14] |
J. Llibre, A survey on the limit cycles of the generalized polynomial Liénard differential equations, Mathematical models in engineering, biology and medicine, AIP Conf. Proc., 1124 (2009), 224-233. |
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