Analytical approximation of cuspidal loops using a nonlinear time transformation method

In this work we consider cuspidal loops, i.e., homoclinic orbits to cuspidal singular points. We develop an iterative procedure, founded on the nonlinear time transformation method, to estimate such codimension-three global bifurcations up to any wanted order, not only in the space of parameters but also in the phase plane. As far as we know, this is the first time in the literature that this theoretical result is achieved for these global connections. The existence and uniqueness of the perturbed solution obtained are proved. To illustrate the effectiveness of the method we study cuspidal loops in two normal forms of degenerate Takens–Bogdanov bifurcations. Excellent agreement is found between our analytical predictions and the corresponding numerical continuations.