May  2018, 17(3): 1305-1316. doi: 10.3934/cpaa.2018063

Cyclicity of degenerate graphic $DF_{2a}$ of Dumortier-Roussarie-Rousseau program

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

Received  October 2016 Revised  March 2017 Published  January 2018

In this paper we finish the study of the cyclicity ( i.e. the maximum number of limit cycles) of the degenerate graphic $DF_{2a}$ of [6] which is initiated in [5]. More precisely, we prove that the graphic $DF_{2a}$ has a finite cyclicity. The goal of the program [6] is to solve the finiteness part of Hilbert's 16th problem for quadratic polynomial systems. We use techniques from geometric singular perturbation theory, including the family blow-up.

Citation: Renato Huzak. Cyclicity of degenerate graphic $DF_{2a}$ of Dumortier-Roussarie-Rousseau program. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1305-1316. doi: 10.3934/cpaa.2018063
References:
[1]

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.  Google Scholar

[2]

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.  Google Scholar

[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.  Google Scholar

[4]

F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100. doi: 10.1090/memo/0577.  Google Scholar

[5]

F. Dumortier and C. Rousseau, Study of the cyclicity of some degenerate graphics inside quadratic systems, Commun. Pure Appl. Anal., 8 (2009), 1133-1157.  doi: 10.3934/cpaa.2009.8.1133.  Google Scholar

[6]

F. DumortierR. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133.  doi: 10.1006/jdeq.1994.1061.  Google Scholar

[7]

R. HuzakP. 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.  Google Scholar

[8]

C. Rousseau, Normal forms, bifurcations and finiteness properties of vector fields, in Normal forms, bifurcations and finiteness problems in differential equations, volume 137 of NATO Sci. Ser. II Math. Phys. Chem., Kluwer Acad. Publ., Dordrecht, (2004), 431-470. doi: 10.1007/978-94-007-1025-2_12.  Google Scholar

[9]

S. Smale, Mathematical problems for the next century, in Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, (2000), 271-294.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

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.  Google Scholar

[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.  Google Scholar

[4]

F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100. doi: 10.1090/memo/0577.  Google Scholar

[5]

F. Dumortier and C. Rousseau, Study of the cyclicity of some degenerate graphics inside quadratic systems, Commun. Pure Appl. Anal., 8 (2009), 1133-1157.  doi: 10.3934/cpaa.2009.8.1133.  Google Scholar

[6]

F. DumortierR. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133.  doi: 10.1006/jdeq.1994.1061.  Google Scholar

[7]

R. HuzakP. 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.  Google Scholar

[8]

C. Rousseau, Normal forms, bifurcations and finiteness properties of vector fields, in Normal forms, bifurcations and finiteness problems in differential equations, volume 137 of NATO Sci. Ser. II Math. Phys. Chem., Kluwer Acad. Publ., Dordrecht, (2004), 431-470. doi: 10.1007/978-94-007-1025-2_12.  Google Scholar

[9]

S. Smale, Mathematical problems for the next century, in Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, (2000), 271-294.  Google Scholar

Figure 1.  The degenerate graphics $DF_{1a}$ ( $b\in]0,2[$ ) and $DF_{2a}$ ( $b=0$ ).
Figure 2.  The degenerate graphic $DF_{2a}$ and the indication of the slow dynamics of (2) for $e_0=e_1=0$ . One can expect limit cycles of (2) to bifurcate from $DF_{2a}$ .
Figure 3.  Six regions covering the sphere in the $(B_0,B_1,B_2)$ -space. Canard limit cycles of (4), Hausdorff-close to $DF_{2a}$ , are only possible for the parameters in the slow-fast Hopf region.
Figure 4.  The transition maps $\Delta_+=\Delta_2\circ \Delta_1$ and $\Delta_-$ .
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