The bifurcation of anongeneric homoclinic orbit (i.e., the orbit comes from theequilibrium along the unstable manifold instead of the center manifold) connecting a nonhyperbolic equilibrium isinvestigated, and the nonhyperbolic equilibrium undergoes apitchfork bifurcation. The existence (resp., nonexistence) of ahomoclinic orbit and an 1-periodic orbit are established when thepitchfork bifurcation does not happen, while as the nonhyperbolicequilibrium undergoes a pitchfork bifurcation, we obtain thesufficient conditions for the existence of homoclinic orbit and twoor three heteroclinic orbits, and so forth. Moreover, we explore thedifference between the bifurcation of the nongeneric homoclinicorbit and the generic one.
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