June  2021, 26(6): 3209-3233. doi: 10.3934/dcdsb.2020225

Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles

1. 

Facultat de Ciències i Tecnologia, Universitat de Vic - Universitat Central de Catalunya (UVic-UCC), 08500 Vic, Spain

2. 

Departamento de Matemàtica, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal

* Corresponding author: Claudia Valls

Received  February 2019 Revised  May 2020 Published  June 2021 Early access  July 2020

Fund Project: The first author is partially supported by the MINECO grant MTM2016-77278-P (FEDER). The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013

We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the $ x- $axis having a nilpotent saddle at the origin.

Citation: Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225
References:
[1]

M. J. ÁlvarezA. Ferragut and X. Jarque, A survey on the blow up technique, Int. J. Bifurcation and Chaos, 21 (2011), 3103-3118.  doi: 10.1142/S0218127411030416.

[2]

J. C. ArtésJ. Llibre and N. Vulpe, Quadratic systems with an integrable saddle: A complete classification in the coefficient space $\mathbb{R}^{12}$, Nonlin. Anal., 75 (2012), 5416-5447.  doi: 10.1016/j.na.2012.04.043.

[3]

N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sb., 30 (1952), 181–196; Amer. Math. Soc. Transl., 100 (1954), 119.

[4]

I. E. ColakJ. Llibre and C. Valls, Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 257 (2014), 1623-1661.  doi: 10.1016/j.jde.2014.05.024.

[5]

I. E. ColakJ. Llibre and C. Valls, Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, Advances in Mathematics, 259 (2014), 655-687.  doi: 10.1016/j.aim.2014.04.002.

[6]

I. E. ColakJ. Llibre and C. Valls, Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. of Differential Equations, 258 (2015), 846-879.  doi: 10.1016/j.jde.2014.10.006.

[7]

I. E. ColakJ. Llibre and C. Valls, Bifurcation diagrams for Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, J. of Differential Equations, 262 (2017), 5518-5533.  doi: 10.1016/j.jde.2017.02.001.

[8]

F. S. DiasJ. Llibre and C. Valls, Polynomial Hamiltonian systems of degree $3$ with symmetric nilpotent centers, Math. Comput. Simulation, 144 (2018), 60-77.  doi: 10.1016/j.matcom.2017.06.002.

[9]

F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, Berlin, 2006.

[10]

H. Dulac, Détermination et integration d'une certaine classe d'equations différentielle ayant par point singulier un centre, Bull. Sci. Math., 32 (1908), 230-252. 

[11]

P. Joyal and C. Rousseau, Saddle quantities and applications, J. Differential Equations, 78 (1999), 374-399.  doi: 10.1016/0022-0396(89)90069-7.

[12]

W. Kapteyn, On the midpoints of integral curves of differential equations of the first Degree, (Dutch) Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl. Nederland, (1911), 1446–1457.

[13]

W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, (Dutch) Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 20 (1912), 1354–1365; Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 21 (1913), 27–33.

[14]

D. SchlomiukJ. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields, Exp. Math., 8 (1990), 673-688. 

[15]

N. I. Vulpe, Affine-invariant conditions for topological distinction of quadratic systems in the presence of a center, Differential Equations, 19 (1983), 371-379. 

show all references

References:
[1]

M. J. ÁlvarezA. Ferragut and X. Jarque, A survey on the blow up technique, Int. J. Bifurcation and Chaos, 21 (2011), 3103-3118.  doi: 10.1142/S0218127411030416.

[2]

J. C. ArtésJ. Llibre and N. Vulpe, Quadratic systems with an integrable saddle: A complete classification in the coefficient space $\mathbb{R}^{12}$, Nonlin. Anal., 75 (2012), 5416-5447.  doi: 10.1016/j.na.2012.04.043.

[3]

N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sb., 30 (1952), 181–196; Amer. Math. Soc. Transl., 100 (1954), 119.

[4]

I. E. ColakJ. Llibre and C. Valls, Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 257 (2014), 1623-1661.  doi: 10.1016/j.jde.2014.05.024.

[5]

I. E. ColakJ. Llibre and C. Valls, Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, Advances in Mathematics, 259 (2014), 655-687.  doi: 10.1016/j.aim.2014.04.002.

[6]

I. E. ColakJ. Llibre and C. Valls, Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. of Differential Equations, 258 (2015), 846-879.  doi: 10.1016/j.jde.2014.10.006.

[7]

I. E. ColakJ. Llibre and C. Valls, Bifurcation diagrams for Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, J. of Differential Equations, 262 (2017), 5518-5533.  doi: 10.1016/j.jde.2017.02.001.

[8]

F. S. DiasJ. Llibre and C. Valls, Polynomial Hamiltonian systems of degree $3$ with symmetric nilpotent centers, Math. Comput. Simulation, 144 (2018), 60-77.  doi: 10.1016/j.matcom.2017.06.002.

[9]

F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, Berlin, 2006.

[10]

H. Dulac, Détermination et integration d'une certaine classe d'equations différentielle ayant par point singulier un centre, Bull. Sci. Math., 32 (1908), 230-252. 

[11]

P. Joyal and C. Rousseau, Saddle quantities and applications, J. Differential Equations, 78 (1999), 374-399.  doi: 10.1016/0022-0396(89)90069-7.

[12]

W. Kapteyn, On the midpoints of integral curves of differential equations of the first Degree, (Dutch) Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl. Nederland, (1911), 1446–1457.

[13]

W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, (Dutch) Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 20 (1912), 1354–1365; Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 21 (1913), 27–33.

[14]

D. SchlomiukJ. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields, Exp. Math., 8 (1990), 673-688. 

[15]

N. I. Vulpe, Affine-invariant conditions for topological distinction of quadratic systems in the presence of a center, Differential Equations, 19 (1983), 371-379. 

Figure 1.  Global phase portraits of Hamiltonian planar polynomial vector field of degree three with a nilpotent saddle at the origin and $ \mathbb{Z}_2 $-symmetric with $ R(x, y) = (x, -y) $. The separatrices are in bold
Figure 2.  Blow up of the origin of $ U_2 $
Figure 3.  The curves where the number of finite singular points can change. The plot of the curves $ b = b_2 $ (continous thick line), $ b = b_3 $ (dashed line), and $ b = a^2 $ (continuous thin line) for $ b>0 $
Figure 4.  The black lines correspond to the curves where the number of finite singular points can change (see Figure 3 for more details), and the gray lines correspond to the curves where the connection of saddles can occur. The connection with the saddle at the origin can occur on the upper gray line, and the connection between two saddles different from the origin can occur on the lower gray line
Figure 5.  Global phase portraits of systems (VII) for $ a = 7 $ and $ b = 34.7 $ (1.17') and $ b = b_2 $ and $ a = 4.5 $ (1.21'). The separatrices are in bold
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