= Global bifurcation
A <global bifurcation> changes an <invariant set> through a global orbit connection or collision that cannot be identified by a single <equilibrium point>'s <linearization> alone. For example, a <global bifurcation> through a saddle <homoclinic orbit> occurs when an outgoing unstable <separatrix> returns to the same <saddle equilibrium>'s <stable manifold>. A smooth global <Poincaré return map> then combines with a local passage map to determine which side contains a <periodic orbit>. A <heteroclinic orbit> instead joins distinct <equilibrium points>; an energy-balance or Melnikov <integral> can locate its breaking in a nearly <Hamiltonian system>.
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