Remarks on homoclinic structure in Bogdanov-Takens map

It is known that in the Bogdanov–Takens map there exists a zone of transversal homoclinic intersections bounded by two curves of homoclinic tangencies. In this paper, we derive an improved asymptotic formula for the homoclinic parameter values of the BT map. We compare two methods to approximate the Bogdanov–Takens map by the time-1 flow of a vector-field, and find that they are equivalent. We show that it is essential to include the second-order terms w.r.t. the parameters to obtain a more accurate asymptotic for the homoclinic zone. We show how to use this new homoclinic asymptotic to compute branches of homoclinic tangencies in the BT map numerically, obtaining the whole homoclinic structure of the Bogdanov–Takens map.