Perturbations of quadratic Hamiltonian systems with symmetry
The main result of the paper is the following Theorem 1. Let \(H\) be a generic cubic Hamiltonian with a centroid symmetry: \(H(- x, - y)= - H(x, y)\). Then any quadratic perturbation of the corresponding Hamiltonian system \(\dot x= H_y+ \varepsilon f(x, y)\), \(\dot y= - H_x+ \varepsilon g(x, y)\) has at most two limit cycles for \(\varepsilon\) small enough. The technique used is based essentially on the notion and on the properties of the centroid curve formed by the mass centres of a continuous family of ovals within the level curves of \(H\).
scientific article; zbMATH DE number 873773
scientific article