Bifurcations and distribution of limit cycles for near-Hamiltonian polynomial systems (Q947543)

The authors are concerned with a perturbed Hamiltonian system \[ \dot{x}=H_{y}+\varepsilon P\left( x,y,\varepsilon,\delta\right) ,\qquad \dot{y}=-H_{x}+\varepsilon Q\left( x,y,\varepsilon,\delta\right) , \] where \(\varepsilon\) is a small nonnegative parameter and \(\delta\in D\subset\mathbb{R}^{p}\) is a vector parameter. They study the relationship between the coefficients of the Melnikov functions \(M_{1},\) \(M_{2},\) \(M_{3}\) and the number of limit cycles for the system and their distribution. As an application, it is shown that the perturbed polynomial system \[ \dot{x}=y,\qquad\dot{y}=-\tfrac{1}{2}x\left( x-1\right) \left( x+1\right) \left( x+2\right) +\varepsilon g\left( x,y\right), \] where \(g(x,y)=a_{01}y+a_{11}xy+a_{21}x^{2}y+a_{03}y^{3}+a_{31}x^{3} y+a_{13}xy^{3},\) can have nine limit cycles with two possible distributions. Note that the unperturbed system has the first integral \[ H\left( x,y\right) =-\tfrac{1}{2}x^{2}-\tfrac{1}{6}x^{3}+\tfrac{1}{4} x^{4}+\tfrac{1}{10}x^{5}+\tfrac{1}{2}y^{2} \] and four singular points, centers \(C_{1}(1,0),\) \(C_{2}(-1,0)\) and saddle points \(S_{1}(0,0),\) \(S_{2}(-2,0).\)