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

Received  January 2017 Revised  August 2017 Published  December 2017

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.

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

[2]

J. C. Artés and J. Llibre, Quadratic Hamiltonian vector fields, J. Differential Equations, 107 (1994), 80-95.  doi: 10.1006/jdeq.1994.1004.  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; Mer. Math. Soc. Transl., 1954 (1954), 1-19.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[7]

I. ColakJ. 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.  Google Scholar

[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.  Google Scholar

[10]

I. Iliev, On second order bifurcations of limit cycles, J. London Math. Soc (2), 58 (1998), 353-366.  doi: 10.1112/S0024610798006486.  Google Scholar

[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

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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.   Google Scholar

[14]

K. E. Malkin, Criteria for the center for a certain differential equation, Vols. Mat. Sb. Vyp., 2 (1964), 87-91.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. M. Peixoto, Dynamical Systems. Proccedings of a Symposium held at the University of Bahia, 389-420, Acad. Press, New York, 1973.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

N. I. Vulpe, Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Differentsial?nye Uravneniya, 19 (1983), 371-379.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  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; Mer. Math. Soc. Transl., 1954 (1954), 1-19.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

I. ColakJ. 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.  Google Scholar

[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.  Google Scholar

[10]

I. Iliev, On second order bifurcations of limit cycles, J. London Math. Soc (2), 58 (1998), 353-366.  doi: 10.1112/S0024610798006486.  Google Scholar

[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.   Google Scholar

[14]

K. E. Malkin, Criteria for the center for a certain differential equation, Vols. Mat. Sb. Vyp., 2 (1964), 87-91.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. M. Peixoto, Dynamical Systems. Proccedings of a Symposium held at the University of Bahia, 389-420, Acad. Press, New York, 1973.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

N. I. Vulpe, Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Differentsial?nye Uravneniya, 19 (1983), 371-379.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Phase portraits for the Hamiltonian systems (2). The separatrices are in bold.
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 3.  Local phase portraits at the equilibria of system (2) if $a = b = 0$ and $c\neq 0$.
Figure 4.  Local phase portrait at the origin of: (a) system (14), (b) system (15)
Figure 5.  Local phase portraits at the equilibria of system (2) if $a = c = 0$ and $b\neq 0$.
Figure 6.  BLocal phase portraits at the origin of systems (18).
Figure 7.  Local phase portraits at the equilibria of system (16) when $a = 0$ and $bc\neq 0$.
Figure 8.  Local phase portrait at the origin of system (21). (a) if $b\geq 0$, (b) if $b<0$.
Figure 9.  Local phase portraits at the equilibria of system (2) when $c = 0$ and $ab\neq0$.
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 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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