Bifurcation of limit cycles for a class of cubic Hamiltonian systems with nesting period annuli

Abstract

In this paper, we obtain the upper bound of the number of zeros of Abelian integral for a class of cubic Hamiltonian systems with nesting period annuli under perturbations of polynomials of degree n. Furthermore, we consider the Hopf and homoclinic bifurcation when a=-1,b=-2,c=1 and n=3, and obtain 18 distributions in which system has at least 3 limit cycles for each case.

Figures (3)

• Figure 1: Phase portraits of system (1.4) for $c>0$. $D_{1}^{+}=\{a<0,b+2a<0,b+2c<0\}$, $D_{2}^{+}=\{a<0,b+2a>0\}$, $l_{1}^{+}=\{a<0,b+2a=0\}$, $D_{3}^{+}=\{a<0,b+2c<0\}$, $l_{2}^{+}=\{a<0,b+2c=0\}$, $D_{4}^{+}=\{a\geq0,b<0,b+2c>0,b^2-4ac>0\}$, $D_{5}^{+}=\{a\geq0,b\geq0\}\cup\{b^2-4ac\leq0,b+2c>0\}$, $l_{3}^{+}=\{b^2-4ac<0,b+2c=0\}$, $D_{6}^{+}=\{b^2-4ac<0,b+2c<0\}$. System has no period annuli for $(a,b)\in\Bbb{R}^2\setminus\{D_{1}^{+}\cup D_{2}^{+}\cup D_{3}^{+}\cup D_{4}^{+}\cup D_{5}^{+}\cup D_{6}^{+}\cup l_{1}^{+}\cup l_{2}^{+}\cup l_{3}^{+}\}$.
• Figure 2: Phase portraits of system (1.4) for $c<0$. $D_{1}^{-}=\{b^2-4ac>0,b+2c>0,b+2a<0\}$, $D_{2}^{-}=\{a<0,b<0\}\cup\{b^2-4ac\leq0,b+2a<0\}$, $l_{1}^{-}=\{b^2-4ac<0,b+2a=0\}$, $D_{3}^{-}=\{b^2-4ac<0,b+2a>0\}$. System has no period annuli for $(a,b)\in\Bbb{R}^2\setminus\{D_{1}^{-}\cup D_{2}^{-}\cup D_{3}^{-}\cup l_{1}^{-}\}$.
• Figure 3: About the calculation of $c_{2,1}$ and $c_{2,2}$: $x_{1}^{*}=x(t_{1})=x(t_{4})$, $x_{2}^{*}=x(t_{2})=x(t_{3})$