Perturbations from a kind of quartic Hamiltonians under general cubic polynomials
The authors extend results by Dumortier and Li on perturbations of a class of quartic Hamiltonian planar vector fields \(X\) (or Liénard equations), admitting now arbitrary cubic perturbations. They prove that if \(X\) has more than one equilibrium, the number of limit cycles of the perturbed system which surround only one center of \(X\) does not exceed twelve. If \(X\) has a saddle loop, they show the existence of a perturbation exhibiting at least three limit cycles, and, moreover, they present a subclass of \(X\) whose perturbations produce at most three limit cycles near the original homoclinic loop. The proofs use standard methods involving Abelian integrals, Picard-Fuchs equations, ratios of (derivatives of) abelian integrals and corresponding Riccati equations. For the expansion of the Abelian integrals near the homoclinic loop, the coefficients up to order four are determined by tedious calculations. As for the zeros of Abelian integrals, the authors apply theorems on the number of intersection points of planar curves that are similar to Khovanskii's results.
scientific article; zbMATH DE number 5643426
scientific article