Abstract
We study bifurcations of two types of homoclinic orbits—a homoclinic orbit with resonant eigenvalues and an inclination-flip homoclinic orbit. For the former, we prove thatN-homoclinic orbits (N⩾3) never bifurcate from the original homoclinic orbit. This answers a problem raised by Chow-Deng-Fiedler (J. Dynam. Diff. Eq. 2, 177–244, 1990). For the latter, we investigate mainlyN-homoclinic orbits andN-periodic orbits forN=1, 2 and determine whether they bifurcate or not under an additional condition on the eigenvalues of the linearized vector field around the equilibrium point.
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Kisaka, M., Kokubu, H. & Oka, H. Bifurcations toN-homoclinic orbits andN-periodic orbits in vector fields. J Dyn Diff Equat 5, 305–357 (1993). https://doi.org/10.1007/BF01053164
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DOI: https://doi.org/10.1007/BF01053164