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 & Continuous Dynamical Systems - A, 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, (2012). Google Scholar

[2]

W. A. Coppel, Some quadratic systems with at most one limit cycle,, in Dynamics reported, (1989), 61. Google Scholar

[3]

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

[4]

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

[5]

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

[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. doi: 10.3934/dcds.2011.29.109. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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. doi: 10.1016/j.jde.2013.01.036. Google Scholar

[15]

R. Huzak, Limit Cycles in Slow-Fast Codimension 3 Bifurcations. Dissertation,, Hasselt University, (2013). Google Scholar

[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. doi: 10.1016/j.jde.2013.07.057. Google Scholar

[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. doi: 10.3934/cpaa.2014.13.2641. Google Scholar

[18]

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

[19]

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

[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., 597 (1977), 335. Google Scholar

[21]

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

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, (2012). Google Scholar

[2]

W. A. Coppel, Some quadratic systems with at most one limit cycle,, in Dynamics reported, (1989), 61. Google Scholar

[3]

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

[4]

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

[5]

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

[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. doi: 10.3934/dcds.2011.29.109. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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. doi: 10.1016/j.jde.2013.01.036. Google Scholar

[15]

R. Huzak, Limit Cycles in Slow-Fast Codimension 3 Bifurcations. Dissertation,, Hasselt University, (2013). Google Scholar

[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. doi: 10.1016/j.jde.2013.07.057. Google Scholar

[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. doi: 10.3934/cpaa.2014.13.2641. Google Scholar

[18]

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

[19]

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

[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., 597 (1977), 335. Google Scholar

[21]

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

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