Global bifurcations of limit cycles for a cubic system (Q2726145)

Consider the Hamiltonian system associated to \(H(x,y)=G(x)+P(y)\) with \(G(x)=\int_0^x u(1+b_1u+b_2u^2) du\) and \(P(y)=\int_0^yu(1+c_1u+c_2u^2) du.\) The authors study the number of limit cycles that bifurcate from their closed trajectories for \(\varepsilon\) small enough, when they consider the special perturbation \(x'=P'(y)-\varepsilon x^2(a_1+a_2x), y'=-G'(x),\) where \(a_1\) and \(a_2\) vary inside a bounded region. The main result obtained is that when \(P''(y)\geq 0\) and the unperturbed system has a center and either two saddle points or no other critical points, then this number of limit cycles is at most one inside any given compact region of the phase plane. The tools used to prove their results are developed in other papers of the second author.