Orbit Flip Homoclinic Loop Resonance Branch

In recent years, the problems about the homoclinic bifurcations and heteroclinic bifurcations have been developed comprehensively (see the references and the references therein), and we get much knowledge of the resonant and nonresonant bifurcations of many kinds of systems. More recently, the study about the homoclinic flip orbit bifurcation has caused the attention of many mathematicians. In [7], the authors deliberate it in great detail and give some beautiful results, especially the nonresonant case. But due to the difficulty encounted, the case with suble resonant eigenvalues, that is ρ1 = λ1 and λ1 = ρ2, is hard to study and the corresponding results are in part conjectural, so it is the most challenging to study. In [16], the authors present a numerical investigation into these bifurcations in a specific three-dimensional vector field, which was constructed by Sandstede's model (see [18]) with three dimensions.In this paper, we study the two kinds of bifurcations of a general four dimensional system in the resonant orbit flip situation, ρ1 =λ1 and λ1 = ρ2, mainly using the method originally established in [23] and [24], then improved in [8]-[10] and [19], etc, that is we first construct the Poincare map to get the associated successor function, finally obtaining the bifurcation function and studying it. We obtain the existence, number, existence regions of the 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit, etc.