Limit cycle bifurcations of some Liénard systems with a cuspidal loop and a homoclinic loop (Q634902)

The authors first establish two general theorems on limit cycle bifurcations of near-Hamiltonian systems near a center, a cuspidal and a homoclinic loop. Then to obtain some lower bounds for the maximal number of limit cycles, they apply the mentioned theorems to study the polynomial Liénard system \[ \dot{x}=y, \;\dot{y}=-g(x) - \varepsilon f(x)y, \] with \(\deg f =n\), \(\deg g =4\) and a small real parameter \(\varepsilon\). New results for \(9\leq n\leq 18\) are presented.