Codimension-3 bifurcations of a class of homoclinic loop with saddle-point (Q2378804)

The paper studies bifurcations of homoclinic loops with double orbit flips (i.e. the orbit connects the strong stable and strong unstable manifolds of a saddle) for four-dimensional systems by the Melnikov method. The existence, numbers, coexistence, and incoexistence of 1-homoclinic orbit, 1-periodic orbit, \(2^n\)-homoclinic orbit, and \(2^n\)-periodic orbit are obtained. Approximate expressions of the corresponding bifurcation surfaces and bifurcation diagrams are presented.