On the shape of limit cycles that bifurcate from Hamiltonian centers (Q1567538)

Consider planar systems of the form \[ \frac{dx}{dt} = \frac{\partial H}{\partial y} (x,y) + \sum^\infty_{k=1} \varepsilon^k f_k (x,y), \;\frac{dy}{dt} = - \frac{\partial H}{\partial x} (x,y) + \sum^\infty_{k=1} \varepsilon^k g_k (x,y),\tag{1} \] where \(H, f_k\) and \(g_k, \;k=1,2\dots\), depend analytically on their variables, \(\varepsilon\) is a small parameter. It is assumed that for \(\varepsilon = 0\), system (1) has a center at the origin. The authors present an explicit method allowing the analytic computation of limit cycles up to any order in \(\varepsilon\) bifurcating from a periodic solution to the Hamiltonian system (1) for \(\varepsilon =0\). The method is based on a result of the authors [Nonlinearity 9, No.~2, 501-516 (1996; Zbl 0886.58087)] that any limit cycle of the system \(dx/dt = P(x,y), \;dy/dt=Q(x,y)\) is contained in \(V^{-1} (0)\) where \(V(x,y)\) is a \(C^1\)-solution to the partial differential equation \(PV_x + QV_y =(P_ x + Q_y)V\). The method is applied to Van der Pol's equation and to a Lienárd system with two limit cycles.