Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop (Q2483361)

There is considered Liénard equations of the form \[ \dot x= y,\quad\dot y= -(x- 2x^3+ x^5)- \varepsilon(\alpha+\beta x^2+ \gamma x^4), \] where \(0<|\varepsilon|\ll 1\), \((\alpha,\beta, \gamma)\in \Lambda\) and \(\Lambda\) is bounded. It is proved that for all \(\alpha\), \(\beta\) the sharp upper bound for the number of zeroes of the Abelian integral \(I(h)\) for \(h\in(0,1/6)\) is \(2\), including multiple zeros.