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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:
 37G20 Hyperbolic singular points with homoclinic trajectories 34C23 Bifurcation (ODE) 34C20 Transformation and reduction of ODE and systems, normal forms 34C37 Homoclinic and heteroclinic solutions of ODE
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