Examples of non-degenerate and degenerate cuspidal loops in planar systems

In this paper we study some examples of non-degenerate and degenerate cuspidal loops in planar systems. A cuspidal loop is a codimension-three homoclinic orbit given by the intersection of the separatrices of an equilibrium of cusp type. Using a Dulac map analysis and asymptotic expansions we study the stability in the neighbourhood of a cusp point. As a first example we consider an enzyme-catalysed reaction model exhibiting a non-degenerate cuspidal loop. All the codimension-one and -two homoclinic bifurcations present in the unfolding of the corresponding cuspidal loop are found in such a realistic model. Finally, the unfolding of a codimension-five Bogdanov-Takens bifurcation is analysed. A degenerate codimension-four cuspidal loop appearing in this system is located on a non-degenerate cuspidal loop curve, and part of the unfolding of such singularity is shown.