Bifurcations of Double Homoclinic Flip Orbits With Resonant Eigenvalues

Concerns double homoclinic loops with or bitflips and two resonant eigenvalues in a fourdimensional system. We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium, and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops, then compose them to get the important Poincar map. A simple calculation gives explicitly an expression of the associated successor function. By a delicate analysis of the bifurcation equation, we obtain the condition that the original double homoclinic loops are kept, and prove the existence and the existence regions of the large 1-homo clinic orbit bifurcation surface, 2-fold large 1-periodic or bit bifurcation surface, large 2-homoclinic or bit bifur cation surface and their appro ximate expressions. We also locate the large periodic orbits and large homoclinic orbits and their number.