January  2016, 36(1): 171-215. doi: 10.3934/dcds.2016.36.171

Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations

1. 

Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, 3590 Diepenbeek

Received  July 2014 Revised  September 2014 Published  June 2015

This paper is the continuation of our previous papers [16] and [17] where we studied small-amplitude limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations. We find optimal upper bounds for the number of small-amplitude limit cycles in these slow-fast codimension 3 bifurcations. We use techniques from geometric singular perturbation theory.
Citation: Renato Huzak. Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 171-215. doi: 10.3934/dcds.2016.36.171
References:
[1]

V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps. Volume 2, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2012, Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous and revised by the authors and James Montaldi, Reprint of the 1988 translation.

[2]

W. A. Coppel, Some quadratic systems with at most one limit cycle, in Dynamics reported, Vol. 2, vol. 2 of Dynam. Report. Ser. Dynam. Systems Appl., Wiley, Chichester, (1989), 61-88.

[3]

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.

[4]

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.

[5]

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.

[6]

P. De Maesschalck and F. 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.

[7]

F. Dumortier, Slow divergence integral and balanced canard solutions, Qual. Theory Dyn. Syst., 10 (2011), 65-85. doi: 10.1007/s12346-011-0038-9.

[8]

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.

[9]

F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields, vol. 1480 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991, Nilpotent singularities and Abelian integrals.

[10]

F. 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.

[11]

F. Dumortier and C. Li, On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations, Nonlinearity, 9 (1996), 1489-1500. doi: 10.1088/0951-7715/9/6/006.

[12]

F. Dumortier and C. Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations, 139 (1997), 41-59. doi: 10.1006/jdeq.1997.3291.

[13]

F. Dumortier and C. Rousseau, Cubic Liénard equations with linear damping, Nonlinearity, 3 (1990), 1015-1039. doi: 10.1088/0951-7715/3/4/004.

[14]

J.-L. Figueras, W. Tucker and J. Villadelprat, Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals, J. Differential Equations, 254 (2013), 3647-3663. doi: 10.1016/j.jde.2013.01.036.

[15]

R. Huzak, Limit Cycles in Slow-Fast Codimension 3 Bifurcations. Dissertation, Hasselt University, Belgium, 2013.

[16]

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.

[17]

R. Huzak, P. De Maesschalck and F. Dumortier, Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations, Communications on Pure and Applied Analysis, 13 (2014), 2641-2673. doi: 10.3934/cpaa.2014.13.2641.

[18]

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

[19]

C. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four, J. Differential Equations, 252 (2012), 3142-3162. doi: 10.1016/j.jde.2011.11.002.

[20]

A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, in Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., Springer, Berlin, 597 (1977), 335-357.

[21]

R. 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.

show all references

References:
[1]

V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps. Volume 2, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2012, Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous and revised by the authors and James Montaldi, Reprint of the 1988 translation.

[2]

W. A. Coppel, Some quadratic systems with at most one limit cycle, in Dynamics reported, Vol. 2, vol. 2 of Dynam. Report. Ser. Dynam. Systems Appl., Wiley, Chichester, (1989), 61-88.

[3]

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.

[4]

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.

[5]

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.

[6]

P. De Maesschalck and F. 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.

[7]

F. Dumortier, Slow divergence integral and balanced canard solutions, Qual. Theory Dyn. Syst., 10 (2011), 65-85. doi: 10.1007/s12346-011-0038-9.

[8]

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.

[9]

F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields, vol. 1480 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991, Nilpotent singularities and Abelian integrals.

[10]

F. 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.

[11]

F. Dumortier and C. Li, On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations, Nonlinearity, 9 (1996), 1489-1500. doi: 10.1088/0951-7715/9/6/006.

[12]

F. Dumortier and C. Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations, 139 (1997), 41-59. doi: 10.1006/jdeq.1997.3291.

[13]

F. Dumortier and C. Rousseau, Cubic Liénard equations with linear damping, Nonlinearity, 3 (1990), 1015-1039. doi: 10.1088/0951-7715/3/4/004.

[14]

J.-L. Figueras, W. Tucker and J. Villadelprat, Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals, J. Differential Equations, 254 (2013), 3647-3663. doi: 10.1016/j.jde.2013.01.036.

[15]

R. Huzak, Limit Cycles in Slow-Fast Codimension 3 Bifurcations. Dissertation, Hasselt University, Belgium, 2013.

[16]

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.

[17]

R. Huzak, P. De Maesschalck and F. Dumortier, Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations, Communications on Pure and Applied Analysis, 13 (2014), 2641-2673. doi: 10.3934/cpaa.2014.13.2641.

[18]

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

[19]

C. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four, J. Differential Equations, 252 (2012), 3142-3162. doi: 10.1016/j.jde.2011.11.002.

[20]

A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, in Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., Springer, Berlin, 597 (1977), 335-357.

[21]

R. 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.

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