# American Institute of Mathematical Sciences

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. [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. [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. [4] F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100. doi: 10.1090/memo/0577. [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. [6] F. Dumortier, R. 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. [7] 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. [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. [9] S. Smale, Mathematical problems for the next century, in Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, (2000), 271-294.

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. [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. [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. [4] F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100. doi: 10.1090/memo/0577. [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. [6] F. Dumortier, R. 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. [7] 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. [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. [9] S. Smale, Mathematical problems for the next century, in Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, (2000), 271-294.
The degenerate graphics $DF_{1a}$ ( $b\in]0,2[$ ) and $DF_{2a}$ ( $b=0$ ).
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 $DF_{2a}$ .
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.
The transition maps $\Delta_+=\Delta_2\circ \Delta_1$ and $\Delta_-$ .
 [1] 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 [2] Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257 [3] Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2/3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621 [4] Ilya Schurov. Duck farming on the two-torus: Multiple canard cycles in generic slow-fast systems. Conference Publications, 2011, 2011 (Special) : 1289-1298. doi: 10.3934/proc.2011.2011.1289 [5] Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233 [6] Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021 [7] Freddy Dumortier, Christiane Rousseau. Study of the cyclicity of some degenerate graphics inside quadratic systems. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1133-1157. doi: 10.3934/cpaa.2009.8.1133 [8] 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 [9] Tetsuya Ishiwata, Shigetoshi Yazaki. A fast blow-up solution and degenerate pinching arising in an anisotropic crystalline motion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2069-2090. doi: 10.3934/dcds.2014.34.2069 [10] Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661 [11] Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123 [12] Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569 [13] Yihong Du, Zongming Guo. The degenerate logistic model and a singularly mixed boundary blow-up problem. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 1-29. doi: 10.3934/dcds.2006.14.1 [14] C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3 (4) : 603-614. doi: 10.3934/mbe.2006.3.603 [15] Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621 [16] Marek Fila, John R. King. Grow up and slow decay in the critical Sobolev case. Networks & Heterogeneous Media, 2012, 7 (4) : 661-671. doi: 10.3934/nhm.2012.7.661 [17] Fabrice Bethuel, Didier Smets. Slow motion for equal depth multiple-well gradient systems: The degenerate case. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 67-87. doi: 10.3934/dcds.2013.33.67 [18] Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025 [19] Monica Marras, Stella Vernier Piro. Blow-up phenomena in reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4001-4014. doi: 10.3934/dcds.2012.32.4001 [20] Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-14. doi: 10.3934/dcdsb.2018069

2016 Impact Factor: 0.801

## Tools

Article outline

Figures and Tables