The twisting bifurcations of double homoclinic loops with resonant eigenvalues (Q369729)

A homoclinic obit is called (non)twisted, if its unstable manifold has an (even) odd number of half-twists along the homoclinic orbit. In the article under review the authors investigate the twisting bifurcations of homoclinic loops in form of figure eight in the space of codimension 3. Roughly speaking, they consider a hyperbolic singularity whose eigenvalues of the linear part has at least a \({1:-1}\) pair. There also exist a homoclinic loop with figure eight and non-degenerated global stable and unstable manifolds. By proper perturbations the complete twisting bifurcation diagram can be drawn, provided that the perturbed system has the special normal form. The whole study goes along the classical way via normal forms, deducing transfer maps and then solving fixed points.NEWLINENEWLINEThe reviewer would like to mention that the normal forms (5) in the paper are not formally proper, although there are closed relationships between normal forms of \({1:-1}\) saddle points and center points.