The uniqueness of bifurcation to separatrix loops in supercritical cases (Q1344125)

The paper considers perturbations of a two-dimensional system having a separatrix loop \(L\) passing through a hyperbolic saddle point. The author studies the existence of a bifurcation of the loop \(L\) to a limit cycle. The author considers the so-called supercritical case. The main theorem gives conditions for the following cases: (i) the perturbed system has a unique stable limit cycle near \(L\), (ii) the perturbed system has a unique unstable cycle near \(L\), (iii) the perturbed system has no limit cycle near \(L\). This result is applied to the study of bifurcations of an example.