Homoclinic behavior around a degenerate heteroclinic cycle in a Lorenz-like system

In this work, we analyze a degenerate heteroclinic cycle that appears in a Lorenz-like system when one of the involved equilibria changes from real saddle to saddle-focus. First, from a theoretical model based on the construction of a Poincaré return map, we demonstrate that an infinite number of homoclinic connections arise from the point of the parameter plane where the degenerate heteroclinic cycle appears. The subsequent numerical study not only illustrates the presence of the first homoclinic orbits in the infinite succession but also allows to find other important local and global organizing centers of codimension two (Bogdanov–Takens bifurcations, degenerate homoclinic and heteroclinic connections, T-points) and three (triple-zero bifurcation, doubly-degenerate heteroclinic cycles, degenerate T-points).