Nongeneric bifurcations near heterodimensional cycles with inclination flip in \(\mathbb{R}^4\)
This paper deals with bifurcations from a heteroclinic cycle in families of vector fields in \(\mathbb{R}^4\). It is assumed that the cycle connects two hyperbolic equilibria \(p_1\) and \(p_2\) with different saddle indices (dimension of the unstable manifold). More precisely, it is assumed that all eigenvalues of the linearizations at the equilibria are real and simple; \(p_1\) has saddle index 3 and \(p_2\) has saddle index 2. The main nongeneric condition is that the (nonrobust) heteroclinic orbit connecting \(p_2\) to \(p_1\) undergoes an inclination flip at \(p_1\). The focus is on bifurcating single around dynamics, i.e., heteroclinic cycles, 1-homoclinic and 1-periodic orbits. To this end, the authors construct a first return map. This construction is based on considerations in [\textit{D. Zhu}, Acta Math. Sin., New Ser. 14, No. 3, 341--352 (1998; Zbl 0932.37032)].
scientific article; zbMATH DE number 5906837