American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020225

Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles

Montserrat Corbera 1, and  Claudia Valls 2,,

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  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.

Keywords: Polynomial Hamiltonian systems, nilpotent saddle, phase portrait, Poincaré compactification.
Mathematics Subject Classification: Primary: 34C05, 34C07; Secondary: 34C08.
Citation: Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020225
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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 Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Blow up of the origin of $ U_2 $
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[2]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[3]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030
[4]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406
[5]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001
[6]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004
[7]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[8]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407
[9]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020368
[10]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176
[11]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173
[12]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
[13]

Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020394
[14]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381
[15]

Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139
[16]

Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010
[17]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021005
[18]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467
[19]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325
[20]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (36)
  • HTML views (195)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences