= Symmetric homoclinic gluing bifurcation
A <symmetric homoclinic gluing bifurcation> joins two symmetry-related <periodic orbits> into one orbit that traverses both lobes of a figure-eight <separatrix>. At the transition, the two loops are <homoclinic orbits> to the same <saddle equilibrium>. The two original cycles spend increasingly long near that saddle before the connection is reached.
A signed transverse coordinate distinguishes the lobes. Reflection can make the corresponding return map odd, so fixed points $x_*$ and $-x_*$ represent the separate cycles, while a two-cycle passing between positive and negative coordinates can represent the glued orbit. The <signed gluing-map reduction> describes the local power-law passage and the global reinjection; its coefficients and saddle eigenvalue ratio determine the existence and stability of these branches.
The geometry alone does not guarantee attracting cycles. For example, the exact Hamiltonian double-well <homoclinic orbits> form a figure-eight but are surrounded by conserved-energy families, rather than isolated attracting cycles. Dissipation, a suitable parameter unfolding and the return-map conditions must supply the gluing scenario. A damped double-well model with symmetry-related Hopf cycles is one setting in which those conditions can be investigated.
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