The non-transverse Shil'nikov-Hopf bifurcation: uncoupling of homoclinic orbits and homoclinic tangencies
The authors consider a degenerate codimension 3 Shilnikov-Hopf bifurcation when at critical values of parameters a three-dimensional smooth vector field has a singular point \(O\) with eigenvalues \((\pm i\omega\), \(\lambda>0)\) and \(O\) has a homoclinic orbit. When changing one of the three parameters on the center manifold of \(O\) it is supposed that the supercritical Andronov-Hopf bifurcation with the birth of stable (on this manifold) periodic orbit occurs. Another parameter is responsible for the splitting of homoclinic orbits. Without an additional degeneration this bifurcation was studied first by L. Belyakov in the framework of two-parameter unfolding. He showed, among other results, that the birth of two primary transverse homoclinic orbits is possible. The authors impose an additional restriction that intersection of stable (more exactly, center-stable) and unstable manifolds along the homoclinic orbit is degenerate with respect to the third parameter. The main point being of interest for the authors is primary periodic orbits near the homoclinic orbit, or, more precisely, how can closed curves on the plane ``period versus parameter'' (unfolding this latter degeneration) form and coalesce with other such curves, which are the bifurcations responsible for these phenomena. The study is based on approximate theoretic first return maps and supported by simulations with a model problem of electronic oscillator.
scientific article; zbMATH DE number 1374344