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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 |
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.
References:
[1] |
M. J. Álvarez, A. 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. Artés, J. 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. Colak, J. 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. Colak, J. 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. Colak, J. 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. Colak, J. 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. Dias, J. 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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[14] |
D. Schlomiuk, J. 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. Álvarez, A. 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. Artés, J. 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. Colak, J. 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. Colak, J. 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. Colak, J. 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. Colak, J. 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. Dias, J. 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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[14] |
D. Schlomiuk, J. 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.
|





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