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January  2011, 29(1): 109-140. doi: 10.3934/dcds.2011.29.109

Detectable canard cycles with singular slow dynamics of any order at the turning point

P. De Maesschalck 1, and  Freddy Dumortier 2,

1. 

Hasselt University, Campus Diepenbeek, Agoralaan-Gebouw D, B-3590 Diepenbeek, Belgium
2. 

Hasselt University, Campus Diepenbeek, Agoralaan gebouw D, B-3590 Diepenbeek, Belgium

Received  January 2010 Revised  April 2010 Published  September 2010

This paper deals with the study of limit cycles that appear in a class of planar slow-fast systems, near a "canard'' limit periodic set of FSTS-type. Limit periodic sets of FSTS-type are closed orbits, composed of a Fast branch, an attracting Slow branch, a Turning point, and a repelling Slow branch. Techniques to bound the number of limit cycles near a FSTS-l.p.s. are based on the study of the so-called slow divergence integral, calculated along the slow branches. In this paper, we extend the technique to the case where the slow dynamics has singularities of any (finite) order that accumulate to the turning point, and in which case the slow divergence integral becomes unbounded. Bounds on the number of limit cycles near the FSTS-l.p.s. are derived by examining appropriate derivatives of the slow divergence integral.
Keywords: singular perturbations, blow-up, slow-fast cycle, generalized Liénard equation., turning point, canards.
Mathematics Subject Classification: Primary: 34E15, 34C26; Secondary: 34E20, 37G1.
Citation: P. De Maesschalck, Freddy Dumortier. Detectable canard cycles with singular slow dynamics of any order at the turning point. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 109-140. doi: 10.3934/dcds.2011.29.109
References:
[1]

P. Bonckaert, Partially hyperbolic fixed points with constraints, Trans. Amer. Math. Soc., 348 (1996), 997-1011. doi: doi:10.1090/S0002-9947-96-01469-9.

[2]

P. Bonckaert, P. De Maesschalck and F. Dumortier, Well adapted normal linearization in singular perturbation problems,, preprint., (). 

[3]

P. De Maesschalck and F. Dumortier, Singular perturbations and vanishing passage through a turning point,, J. Differential Equations, ().  doi: doi:10.1016/j.jde.2009.11.009.

[4]

P. De Maesschalck and F. Dumortier, Slow-fast Bogdanov-Takens bifurcations,, J. Differential Equations, ().  doi: doi:10.1016/j.jde.2010.07.022.

[5]

P. De Maesschalck and F. Dumortier, Time analysis and entry-exit relation near planar turning points, J. Differential Equations, 215 (2005), 225-267. doi: doi:10.1016/j.jde.2005.01.004.

[6]

P. De Maesschalck and F. Dumortier, Canard solutions at non-generic turning points, Trans. Amer. Math. Soc., 358 (2006), 2291-2334. doi: doi:10.1090/S0002-9947-05-03839-0.

[7]

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: doi:10.1017/S0308210506000199.

[8]

F. Dumortier, Slow divergence integral and balanced canard solutions,, Qualitative Theory and Dynamical Systems, (). 

[9]

F. Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations, 224 (2006), 296-313. doi: doi:10.1016/j.jde.2005.08.011.

[10]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368. doi: doi:10.1006/jdeq.2000.3929.

[11]

D. Panazzolo, Desingularization of nilpotent singularities in families of planar vector fields, Mem. Amer. Math. Soc., 158 (2002).

[12]

R. Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst., 17 (2007), 441-448. doi: doi:10.3934/dcds.2007.17.441.

show all references

References:
[1]

P. Bonckaert, Partially hyperbolic fixed points with constraints, Trans. Amer. Math. Soc., 348 (1996), 997-1011. doi: doi:10.1090/S0002-9947-96-01469-9.

[2]

P. Bonckaert, P. De Maesschalck and F. Dumortier, Well adapted normal linearization in singular perturbation problems,, preprint., (). 

[3]

P. De Maesschalck and F. Dumortier, Singular perturbations and vanishing passage through a turning point,, J. Differential Equations, ().  doi: doi:10.1016/j.jde.2009.11.009.

[4]

P. De Maesschalck and F. Dumortier, Slow-fast Bogdanov-Takens bifurcations,, J. Differential Equations, ().  doi: doi:10.1016/j.jde.2010.07.022.

[5]

P. De Maesschalck and F. Dumortier, Time analysis and entry-exit relation near planar turning points, J. Differential Equations, 215 (2005), 225-267. doi: doi:10.1016/j.jde.2005.01.004.

[6]

P. De Maesschalck and F. Dumortier, Canard solutions at non-generic turning points, Trans. Amer. Math. Soc., 358 (2006), 2291-2334. doi: doi:10.1090/S0002-9947-05-03839-0.

[7]

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: doi:10.1017/S0308210506000199.

[8]

F. Dumortier, Slow divergence integral and balanced canard solutions,, Qualitative Theory and Dynamical Systems, (). 

[9]

F. Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations, 224 (2006), 296-313. doi: doi:10.1016/j.jde.2005.08.011.

[10]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368. doi: doi:10.1006/jdeq.2000.3929.

[11]

D. Panazzolo, Desingularization of nilpotent singularities in families of planar vector fields, Mem. Amer. Math. Soc., 158 (2002).

[12]

R. Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst., 17 (2007), 441-448. doi: doi:10.3934/dcds.2007.17.441.

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