Bifurcations of Generic heteroclinic Loop Accompanied by Transcritical bifurcation

The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1 and one hyperbolic saddle p2 are investigated, where p1 is assumed to undergo transcritical bifurcation. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p1 (resp. p2) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p1 splits into two hyperbolic saddles p1 and p1, a heteroclinic loop connecting p1 and p2, homoclinic loop with p 1 1 (resp. p2) and heteroclinic orbit joining p 0 1 and p1 (resp. p1 and p2; p2 and p1) are found. The results achieved here can be extended to higher dimensional systems.