# American Institute of Mathematical Sciences

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  March 2018 Early access  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
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##### References:
Phase portraits for the Hamiltonian systems (2). The separatrices are in bold.
The blow-ups of the origin of the chart $U_1$ for system (8). The dotted line represents a straight line of equilibria.
Local phase portraits at the equilibria of system (2) if $a = b = 0$ and $c\neq 0$.
Local phase portrait at the origin of: (a) system (14), (b) system (15)
Local phase portraits at the equilibria of system (2) if $a = c = 0$ and $b\neq 0$.
BLocal phase portraits at the origin of systems (18).
Local phase portraits at the equilibria of system (16) when $a = 0$ and $bc\neq 0$.
Local phase portrait at the origin of system (21). (a) if $b\geq 0$, (b) if $b<0$.
Local phase portraits at the equilibria of system (2) when $c = 0$ and $ab\neq0$.
Local phase portraits at the equilibria $p_2$ and $p_3$ of system (23) after translating to the origin. (a) $p_2$, (b) $p_3$
Local phase portraits at the equilibria of system associated to Hamiltonian (3) when $ac \neq 0$.
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 $c>b^2/4$.
, 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).
(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 $f_{35}>0$.
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 $h_5<h_3<h_2$,
(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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