Abstract
We study a bifurcation of homoclinic and heteroclinic orbits in a two or more parameter family of autonomous ODEs, where the unperturbed system has two heteroclinic orbits joined at a common saddle point. Under some assumptions on eigenvalues of the linearized equation at equilibrium points and on a non-degeneracy condition for the system, we can show that heteroclinic orbits of new type appear besides the persistent ones of the unperturbed system. A bifurcation diagram is given for such families. Some homoclinic bifurcations are also treated including the one producing a twice-rounding homoclinic orbit.
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Kokubu, H. Homoclinic and heteroclinic bifurcations of Vector fields. Japan J. Appl. Math. 5, 455–501 (1988). https://doi.org/10.1007/BF03167912
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DOI: https://doi.org/10.1007/BF03167912