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.

For a symmetric real saddle with one unstable direction and a strongly contracting transverse direction, composing the power-law local passage with a smooth global return gives this leading Poincaré return map, with $A>0$ and saddle index $\delta$. It is undefined at the stable-manifold crossing $x=0$. For $\delta >1$, small fixed points on each branch are attracting for $\mu <0$; for $\mu >0$ an attracting alternating-sign period-two orbit satisfies $s=\mu -A{s}^{\delta }$, giving gluing of two one-lobe flow cycles into one two-lobe cycle. For $\delta <1$ and small $\mu >0$, on $I=[-c\mu ,c\mu ]$, $0<c<1$, the inverse branchesmap into $I$ and contract. Infinite branch choices define an invariant Cantor set with full two-symbol dynamics. In the two-dimensional return, transverse contraction turns this mechanism into a Smale horseshoe. An attracting Lorenz set additionally requires appropriate trapping and global-return properties.

For $x^{\prime }=-\mu \mathrm{sgn}(x)+A\mathrm{sgn}(y)|x{|}^{\delta }$, $y^{\prime }=\mathrm{sgn}(x)$, the invariant sign sheet has $y=\pm 1$ after one iterate. There $z=-xy$ obeys the displayed one-dimensional map. A negative fixed $z$ lifts to two symmetry-related fixed points, while a positive fixed $z$ lifts to one period-two orbit. The origin of the return map represents the separatrix, where the flow return time diverges, and is not an ordinary finite-period orbit of a group action.

The fixed point relation is $\mu =z-A\mathrm{sgn}(z)|z{|}^{\delta }$. Setting the fixed-point multiplier $A\delta |z{|}^{\delta -1}$ equal to one gives the two saddle-node bifurcations at $z=\pm r$. For fixed $0<A<1$, as $\delta \uparrow 1$ from below their parameter width tends to zero as $(1-\delta )A^{1/(1-\delta )}/e$, forming an exponentially narrow gluing-map cusp. Three scalar fixed points coexist inside this region: two stable outer branches and one unstable inner branch. For $\delta >1$ near one the formal saddle-node bifurcations recede to infinity, outside the local gluing-map domain.

For fixed $0<A<1$ and $\delta <1$, the two folds of the signed gluing-map reduction enclose three scalar fixed-point branches. Writing $\delta =1-\eta$ gives width $C=\eta (1-\eta {)}^{-1}[A(1-\eta ){]}^{1/\eta }$. Since $(1-\eta {)}^{1/\eta }\to e^{-1}$, the displayed essential asymptotic follows. The fold curves meet with exponentially small width; this is not the ordinary algebraic cusp normal form of a smooth cubic vector field. A negative fixed point lifts to two symmetry-related flow cycles, so the cusp describes distinct return-map branches rather than counting every physical cycle separately.