Bifurcations of limit cycles from cubic Hamiltonian systems with a center and a homoclinic saddle-loop
This paper is concerned with the maximum number of limit cycles of the system \[ \dot x= y,\quad\dot y= kx- (k+1) x^2+ x^3+ \varepsilon(\alpha+ \beta x+\gamma x^2) y, \] when \(k> {11+ \sqrt{33}\over 4}\), \(0< |\varepsilon|\ll 1\), \(|\alpha|+|\beta|+ |\gamma|> 0\). Such a problem is related to the weak form of Hilbert's 16th problem, as proposed by Arnold. The system is considered as a perturbation of the Hamiltonian system obtained for \(\varepsilon= 0\). The authors estimate the number of zeroes of a suitable Abelian integral and prove that for small values of \(\varepsilon\) there exist no more than 2 limit cycles. This extends previous results about limit cycles of perturbations of cubic Hamiltonian systems.
scientific article; zbMATH DE number 1505972