Principal Poincaré-Pontryagin function of polynomial perturbations of the Hamiltonian triangle
The author investigates the displacement map \(D(t,\varepsilon)\) for a small polynomial perturbation \[ dF(x,y)+\varepsilon[P_d(x,y)dx+Q_d(x,y)dy]=0, \] where \(F(x,y)=x[y^2-(x-3)^2]\) is the so called Hamiltonian triangle and \(P_d,Q_d\) are polynomials of degree \(n\). The first nonzero coefficient \(M_k(t)\) in the expansion with respect to the small parameter, \[ D(t,\varepsilon)=\varepsilon^k M_k(t)+O(\varepsilon^{k+1}), \] is sometimes called Poincaré-Pontryagin function and its zeroes correspond to the limit cycles that tend as \(\varepsilon\to 0\) to periodic orbits of the unperturbed system. The number \(k\) depends on the perturbation. In the quadratic case \(d=2\), the function \(D(t,\varepsilon)\) was first studied by \textit{H. Żołądek} [J. Differ. Equations 109, 223--273 (1994; Zbl 0797.34044)] and by \textit{I. D. Iliev} [Adv. Differ. Equ. 1, No. 4, 689--708 (1996; Zbl 0851.34042), J. Differ. Equations 128, No. 1, 309--326 (1996; Zbl 0853.58084)]. It was established that \(M_1\) and \(M_2\) are Abelian integrals but \(M_3\) is not; moreover, \(M_1(t)\neq 0\), \(M_2(t)\) has at most one zero, \(M_3\) and \(D(t,\varepsilon)\) have at most three zeroes. In this paper the author proves for any \(d\) that the first nonzero term \(M_k(t)\) belongs to a \({\mathbf C}[t,1/t]\) module of the same three integrals over \(F=t\) as those included in \(M_3(t)\) when \(d=2\). Also, for \(d\geq 5\), the coefficient \(M_2(t)\) is not (generically) an Abelian integral, too.
scientific article; zbMATH DE number 5156804