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November  2014, 13(6): 2641-2673. doi: 10.3934/cpaa.2014.13.2641

Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations

Renato Huzak 1, , P. De Maesschalck 2, and  Freddy Dumortier 3,

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

Received  October 2013 Revised  April 2014 Published  July 2014

In this paper we continue the study of ``large" small-amplitude limit cycles in slow-fast codimension 3 elliptic bifurcations which is initiated in [8]. Our treatment is based on blow-up and good normal forms.
Keywords: cyclicity, Blow-up, center manifold, normal form, singular perturbations.
Mathematics Subject Classification: Primary: 34E15, 37G15, 34E20, 34C23, 34C2.
Citation: Renato Huzak, P. De Maesschalck, Freddy Dumortier. Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2641-2673. doi: 10.3934/cpaa.2014.13.2641
References:
[1]

P. De Maesschalck and F. Dumortier, Slow-fast Bogdanov-Takens bifurcations,, \emph{J. Differential Equations}, 250 (2011), 1000.  doi: 10.1016/j.jde.2010.07.022.  Google Scholar

[2]

F. Dumortier and R. Roussarie, Birth of canard cycles,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 723.  doi: 10.3934/dcdss.2009.2.723.  Google Scholar

[3]

P. De Maesschalck and F. Dumortier, Singular perturbations and vanishing passage through a turning point,, \emph{J. Differential Equations}, 248 (2010), 2294.  doi: 10.1016/j.jde.2009.11.009.  Google Scholar

[4]

P. De Maesschalck and F. Dumortier, Canard cycles in the presence of slow dynamics with singularities,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 138 (2008), 265.  doi: 10.1017/S0308210506000199.  Google Scholar

[5]

F. Dumortier, R. Roussarie, J. Sotomayor, and H. Zoladek, Bifurcations of Planar Vector Fields,, Lecture Notes in Mathematics, 1480 (1991).   Google Scholar

[6]

Freddy Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations,, \emph{J. Differential Equations}, 224 (2006), 296.  doi: 10.1016/j.jde.2005.08.011.  Google Scholar

[7]

Robert Roussarie, Putting a boundary to the space of Liénard equations,, \emph{Discrete Contin. Dyn. Syst.}, 17 (2007), 441.  doi: 10.3934/dcds.2007.17.441.  Google Scholar

[8]

R. Huzak, P. De Maesschalck and F. Dumortier, Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations,, \emph{J. Differential Equations}, 255 (2013), 4012.  doi: 10.1016/j.jde.2013.07.057.  Google Scholar

[9]

Peter De Maesschalck and Freddy Dumortier, Detectable canard cycles with singular slow dynamics of any order at the turning point,, \emph{Discrete Contin. Dyn. Syst.}, 29 (2011), 109.  doi: 10.3934/dcds.2011.29.109.  Google Scholar

[10]

P. De Maesschalck and F. Dumortier, Time analysis and entry-exit relation near planar turning points,, \emph{J. Differential Equations}, 215 (2005), 225.  doi: 10.1016/j.jde.2005.01.004.  Google Scholar

[11]

P. Bonckaert, P. De Maesschalck and F. Dumortier, Well adapted normal linearization in singular perturbation problems,, \emph{J. Dynam. Differential Equations}, 23 (2011), 115.  doi: 10.1007/s10884-010-9191-0.  Google Scholar

[12]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion,, \emph{J. Differential Equations}, 174 (2001), 312.  doi: 10.1006/jdeq.2000.3929.  Google Scholar

[13]

Daniel Panazzolo, Desingularization of nilpotent singularities in families of planar vector fields,, \emph{Mem. Amer. Math. Soc.}, 158 (2002).  doi: 10.1090/memo/0753.  Google Scholar

[14]

J. Llibre, A survey on the limit cycles of the generalized polynomial Liénard differential equations,, \emph{Mathematical models in engineering, 1124 (2009), 224.   Google Scholar

show all references

References:
[1]

P. De Maesschalck and F. Dumortier, Slow-fast Bogdanov-Takens bifurcations,, \emph{J. Differential Equations}, 250 (2011), 1000.  doi: 10.1016/j.jde.2010.07.022.  Google Scholar

[2]

F. Dumortier and R. Roussarie, Birth of canard cycles,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 2 (2009), 723.  doi: 10.3934/dcdss.2009.2.723.  Google Scholar

[3]

P. De Maesschalck and F. Dumortier, Singular perturbations and vanishing passage through a turning point,, \emph{J. Differential Equations}, 248 (2010), 2294.  doi: 10.1016/j.jde.2009.11.009.  Google Scholar

[4]

P. De Maesschalck and F. Dumortier, Canard cycles in the presence of slow dynamics with singularities,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 138 (2008), 265.  doi: 10.1017/S0308210506000199.  Google Scholar

[5]

F. Dumortier, R. Roussarie, J. Sotomayor, and H. Zoladek, Bifurcations of Planar Vector Fields,, Lecture Notes in Mathematics, 1480 (1991).   Google Scholar

[6]

Freddy Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations,, \emph{J. Differential Equations}, 224 (2006), 296.  doi: 10.1016/j.jde.2005.08.011.  Google Scholar

[7]

Robert Roussarie, Putting a boundary to the space of Liénard equations,, \emph{Discrete Contin. Dyn. Syst.}, 17 (2007), 441.  doi: 10.3934/dcds.2007.17.441.  Google Scholar

[8]

R. Huzak, P. De Maesschalck and F. Dumortier, Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations,, \emph{J. Differential Equations}, 255 (2013), 4012.  doi: 10.1016/j.jde.2013.07.057.  Google Scholar

[9]

Peter De Maesschalck and Freddy Dumortier, Detectable canard cycles with singular slow dynamics of any order at the turning point,, \emph{Discrete Contin. Dyn. Syst.}, 29 (2011), 109.  doi: 10.3934/dcds.2011.29.109.  Google Scholar

[10]

P. De Maesschalck and F. Dumortier, Time analysis and entry-exit relation near planar turning points,, \emph{J. Differential Equations}, 215 (2005), 225.  doi: 10.1016/j.jde.2005.01.004.  Google Scholar

[11]

P. Bonckaert, P. De Maesschalck and F. Dumortier, Well adapted normal linearization in singular perturbation problems,, \emph{J. Dynam. Differential Equations}, 23 (2011), 115.  doi: 10.1007/s10884-010-9191-0.  Google Scholar

[12]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion,, \emph{J. Differential Equations}, 174 (2001), 312.  doi: 10.1006/jdeq.2000.3929.  Google Scholar

[13]

Daniel Panazzolo, Desingularization of nilpotent singularities in families of planar vector fields,, \emph{Mem. Amer. Math. Soc.}, 158 (2002).  doi: 10.1090/memo/0753.  Google Scholar

[14]

J. Llibre, A survey on the limit cycles of the generalized polynomial Liénard differential equations,, \emph{Mathematical models in engineering, 1124 (2009), 224.   Google Scholar

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