Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis

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 $f_{35}>0$.