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Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips. (English) Zbl 1170.37325
Summary: Homoclinic bifurcations in four-dimensional vector fields are investigated by setting up a local coordinate near a homoclinic orbit. This homoclinic orbit is principal but its stable and unstable foliations take inclination flip. The existence, nonexistence, and uniqueness of the 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold 1-periodic orbit and three-fold 1-periodic orbit are also obtained. It is indicated that the number of periodic orbits bifurcated from this kind of homoclinic orbits depends heavily on the strength of the inclination flip.
MSC:
37G20Hyperbolic singular points with homoclinic trajectories
34C23Bifurcation (ODE)
34C20Transformation and reduction of ODE and systems, normal forms
34C37Homoclinic and heteroclinic solutions of ODE