 Symmetric homoclinic gluing bifurcation

Two symmetry-related periodic orbits can meet the two loops of a symmetric saddle's figure-eight separatrix. On the other side a single periodic orbit can traverse both lobes. The saddle value and nonlinear return-map coefficients determine stability. In a damped double-well oscillator the symmetry-related stable cycles born at the two well Hopf bifurcations provide a natural setting for this merger.