Figure 1.
Phase portraits for the Hamiltonian systems (2). The separatrices are in bold.
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March
2018, 23(2): 887-912.
doi: 10.3934/dcdsb.2018047
Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis
| 1. | Departament de Matemàtiques, Facultat de Ciències Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain |
| 2. | Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Ⅷ-Región, Chile |
| 3. | Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Ⅷ-región, Chile |
We provide the phase portraits in the Poincaré disk for all the linear type centers of polynomial Hamiltonian systems with nonlinearities of degree $4$ symmetric with respect to the $y$-axis given by the Hamiltonian function $H(x,y) =1/2(x^2+y^2)+ax^4y+bx^2y^3+cy^5$ in function of its parameters.
Keywords:
Hamiltonian systems,
linear type centers,
quartic polynomial,
polynomial vector fields,
phase portraits.
Mathematics Subject Classification:
Primary 34C07; Secondary: 34C08.
Citation:
Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis. Discrete & Continuous Dynamical Systems - B,
2018, 23
(2)
: 887-912.
doi: 10.3934/dcdsb.2018047
![]()
References:
| [1] |
V. I. Arnold and Y. S. Ilyashenko, Dynamical Systems I, Ordinary Differential Equations. Encyclopaedia of Mathematical Sciences, Vols 1-2, Springer-Verlag, Heidelberg, 1988. Google Scholar |
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J. C. Artés and J. Llibre,
Quadratic Hamiltonian vector fields, J. Differential Equations, 107 (1994), 80-95.
doi: 10.1006/jdeq.1994.1004. |
| [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; Mer. Math. Soc. Transl., 1954 (1954), 1-19. |
| [4] |
J. Chavarriga and J. Giné,
Integrability of a linear center perturbed by a fourth degree homogeneous polynomial, Publ. Mat., 40 (1996), 21-39.
doi: 10.5565/PUBLMAT_40196_03. |
| [5] |
J. Chavarriga and J. Giné,
Integrability of a linear center perturbed by a fifth degree homogeneous polynomial, Publ. Mat., 41 (1997), 335-356.
doi: 10.5565/PUBLMAT_41297_02. |
| [6] |
A. Cima and J. Llibre,
Algebraic and topological classification of the homogeneous cubic vector fields in the plane, J. of Math. Anal. and Appl., 147 (1990), 420-448.
doi: 10.1016/0022-247X(90)90359-N. |
| [7] |
I. Colak, J. Llibre and C. Valls,
Hamiltonian non-degenerate centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 257 (2014), 1623-1661.
doi: 10.1016/j.jde.2014.05.024. |
| [8] |
H. Dulac, Détermination et integration d' une certaine classe d' équations différentielle ayant par point singulier un centre, Bull. Sci. Math. Sér.(2), 32 (1908), 230-252. Google Scholar |
| [9] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Spring-Verlag, 2006. |
| [10] |
I. Iliev,
On second order bifurcations of limit cycles, J. London Math. Soc (2), 58 (1998), 353-366.
doi: 10.1112/S0024610798006486. |
| [11] |
W. Kapteyn, On the midpoints of integral curves of differential equations of the first Degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl. Nederland, 19 (1911), 1446-1457. Google Scholar |
| [12] |
W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 20 (1912), 1354-1365; Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 21 (1913), 27-33 (in Dutch). Google Scholar |
| [13] |
J. Llibre and A. C. Mereu,
Limit cycles for a class of discontinuous generalized Lienard polynomial differential equations, Electronic J. of Differential Equations, 2013 (2013), 1-8.
|
| [14] |
K. E. Malkin,
Criteria for the center for a certain differential equation, Vols. Mat. Sb. Vyp., 2 (1964), 87-91.
|
| [15] |
L. Markus,
Global structure of ordinary differential equations in the plane, Trans. Amer. Math Soc., 76 (1954), 127-148.
doi: 10.1090/S0002-9947-1954-0060657-0. |
| [16] |
D. A. Neumann,
Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73-81.
doi: 10.1090/S0002-9939-1975-0356138-6. |
| [17] |
M. M. Peixoto, Dynamical Systems. Proccedings of a Symposium held at the University of Bahia, 389-420, Acad. Press, New York, 1973. |
| [18] |
H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Journal de Mathématiques, 37 (1881), 375-422; Oeuvres de Henri Poincaré, Gauthier-Villars, Paris, 1 (1951), 3-84. Google Scholar |
| [19] |
C. Rousseau and D. Schlomiuk,
Cubic vector fields symmetric with respect to a center, J. Differential Equations, 123 (1995), 388-436.
doi: 10.1006/jdeq.1995.1168. |
| [20] |
D. Schlomiuk,
Algebraic particular integrals, integrability and the problem of the centre, Trans. Amer. Math. Soc., 338 (1993), 799-841.
doi: 10.1090/S0002-9947-1993-1106193-6. |
| [21] |
N. I. Vulpe,
Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Differentsial?nye Uravneniya, 19 (1983), 371-379.
|
| [22] |
N. I. Vulpe and K. S. Sibirskii, Centro-affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities, Dokl. Akad. Nauk. SSSR, 301 (1988), 1297-1301 (in Russian); translation in: Soviet Math. Dokl., 38 (1989), 198-201. |
| [23] |
H. Żołądek,
The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal., 4 (1994), 79-136.
doi: 10.12775/TMNA.1994.024. |
| [24] |
H. Żołądek, Remarks on: 'The classification of reversible cubic systems with center', Topol. Methods Nonlinear Anal., 4 (1994), 79-136], Topol. Methods Nonlinear Anal., 8 (1996), 335-342. |
show all references
References:
| [1] |
V. I. Arnold and Y. S. Ilyashenko, Dynamical Systems I, Ordinary Differential Equations. Encyclopaedia of Mathematical Sciences, Vols 1-2, Springer-Verlag, Heidelberg, 1988. Google Scholar |
| [2] |
J. C. Artés and J. Llibre,
Quadratic Hamiltonian vector fields, J. Differential Equations, 107 (1994), 80-95.
doi: 10.1006/jdeq.1994.1004. |
| [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; Mer. Math. Soc. Transl., 1954 (1954), 1-19. |
| [4] |
J. Chavarriga and J. Giné,
Integrability of a linear center perturbed by a fourth degree homogeneous polynomial, Publ. Mat., 40 (1996), 21-39.
doi: 10.5565/PUBLMAT_40196_03. |
| [5] |
J. Chavarriga and J. Giné,
Integrability of a linear center perturbed by a fifth degree homogeneous polynomial, Publ. Mat., 41 (1997), 335-356.
doi: 10.5565/PUBLMAT_41297_02. |
| [6] |
A. Cima and J. Llibre,
Algebraic and topological classification of the homogeneous cubic vector fields in the plane, J. of Math. Anal. and Appl., 147 (1990), 420-448.
doi: 10.1016/0022-247X(90)90359-N. |
| [7] |
I. Colak, J. Llibre and C. Valls,
Hamiltonian non-degenerate centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 257 (2014), 1623-1661.
doi: 10.1016/j.jde.2014.05.024. |
| [8] |
H. Dulac, Détermination et integration d' une certaine classe d' équations différentielle ayant par point singulier un centre, Bull. Sci. Math. Sér.(2), 32 (1908), 230-252. Google Scholar |
| [9] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Spring-Verlag, 2006. |
| [10] |
I. Iliev,
On second order bifurcations of limit cycles, J. London Math. Soc (2), 58 (1998), 353-366.
doi: 10.1112/S0024610798006486. |
| [11] |
W. Kapteyn, On the midpoints of integral curves of differential equations of the first Degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl. Nederland, 19 (1911), 1446-1457. Google Scholar |
| [12] |
W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 20 (1912), 1354-1365; Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 21 (1913), 27-33 (in Dutch). Google Scholar |
| [13] |
J. Llibre and A. C. Mereu,
Limit cycles for a class of discontinuous generalized Lienard polynomial differential equations, Electronic J. of Differential Equations, 2013 (2013), 1-8.
|
| [14] |
K. E. Malkin,
Criteria for the center for a certain differential equation, Vols. Mat. Sb. Vyp., 2 (1964), 87-91.
|
| [15] |
L. Markus,
Global structure of ordinary differential equations in the plane, Trans. Amer. Math Soc., 76 (1954), 127-148.
doi: 10.1090/S0002-9947-1954-0060657-0. |
| [16] |
D. A. Neumann,
Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73-81.
doi: 10.1090/S0002-9939-1975-0356138-6. |
| [17] |
M. M. Peixoto, Dynamical Systems. Proccedings of a Symposium held at the University of Bahia, 389-420, Acad. Press, New York, 1973. |
| [18] |
H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Journal de Mathématiques, 37 (1881), 375-422; Oeuvres de Henri Poincaré, Gauthier-Villars, Paris, 1 (1951), 3-84. Google Scholar |
| [19] |
C. Rousseau and D. Schlomiuk,
Cubic vector fields symmetric with respect to a center, J. Differential Equations, 123 (1995), 388-436.
doi: 10.1006/jdeq.1995.1168. |
| [20] |
D. Schlomiuk,
Algebraic particular integrals, integrability and the problem of the centre, Trans. Amer. Math. Soc., 338 (1993), 799-841.
doi: 10.1090/S0002-9947-1993-1106193-6. |
| [21] |
N. I. Vulpe,
Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Differentsial?nye Uravneniya, 19 (1983), 371-379.
|
| [22] |
N. I. Vulpe and K. S. Sibirskii, Centro-affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities, Dokl. Akad. Nauk. SSSR, 301 (1988), 1297-1301 (in Russian); translation in: Soviet Math. Dokl., 38 (1989), 198-201. |
| [23] |
H. Żołądek,
The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal., 4 (1994), 79-136.
doi: 10.12775/TMNA.1994.024. |
| [24] |
H. Żołądek, Remarks on: 'The classification of reversible cubic systems with center', Topol. Methods Nonlinear Anal., 4 (1994), 79-136], Topol. Methods Nonlinear Anal., 8 (1996), 335-342. |
Figure 2.
The blow-ups of the origin of the chart $U_1$ for system (8). The dotted line represents a straight line of equilibria.
Figure 4.
Local phase portrait at the origin of: (a) system (14), (b) system (15)
Figure 6.
BLocal phase portraits at the origin of systems (18).
Figure 10.
Local phase portraits at the equilibria $p_2$ and $p_3$ of system (23) after translating to the origin. (a) $p_2$ , (b) $p_3$
Figure 11.
Local phase portraits at the equilibria of system associated to Hamiltonian (3) when $ac \neq 0$ .
Figure 12.
Graph of the function $f(b,c) = h_{2}-h_{5}$ on the $(b,c)$ -plan. In cases (a): $b^2-4c<0$ , $\Delta >0 $ , $0\leq b<4/3$ and $c>2b/5$ , (b): $b^2-4c<0$ , $\Delta>0 $ , $b\leq 0$ and $c>b^2/4$ .
Figure 11(g), i.e., when (ⅶ) holds, (b): Graph of the functions $f(b,c) = h_{3}-h_{5} $ and its intersection with the $(b,c)$ -plane under the conditions of the existence of Figure 11(f), i.e., in the case (ⅵ).">Figure 13.
(a): Graph of the functions $f(b,c) = h_{2}-h_{5} $ and its intersection with the $(b,c)$ -plane, under the conditions of the existence of Figure 11(g), i.e., when (ⅶ) holds, (b): Graph of the functions $f(b,c) = h_{3}-h_{5} $ and its intersection with the $(b,c)$ -plane under the conditions of the existence of Figure 11(f), i.e., in the case (ⅵ).
Figure 14.
(a): Graph of the functions $f_{35}(b,c) = h_{3}-h_{5} $ and its intersection with the $(b,c)$ -plane, i.e., when (ⅷ) holds, (b): Graph of the functions $f_{23}(b,c)$ and $f_{25}$ in the region where $f_{35}>0$ .
Figure 15.
Level curve $h_2$ passing though $e_2$ . (a) Region $h_5<h_2<h_3$ , (b) Region $h_5<h_2 = h_3$ , (c) Region $h_5<h_3<h_2$ ,
Figure 16.
(a): Graph of the functions $f_{35}(b,c) = 0$ , (b): Graph of the functions $f_{23}(b,c) = 0$ and $f_{25}(b,c) = 0$ in the region where $f_{35}>0$ .
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