 Asymmetric planar gluing return map

Near two homoclinic orbits of a planar saddle equilibrium with contracting ratio $\delta >1$, the signed Poincaré return map has separate offsets $\mu ,\nu$. Two single-lobe periodic orbits exist for $\mu <0$ and $\nu <0$, respectively. A two-lobe periodic orbit is represented by positive $p,q$ solving the displayed equations. Its boundary consists of the compound homoclinic orbit curves $\mu =-A{\nu }^{\delta }$ for $\nu >0$ and $\nu =-B{\mu }^{\delta }$ for $\mu >0$, with $A,B>0$. In a sufficiently small return domain the derivatives tend to zero, so each admissible symbolic branch has at most one attracting periodic orbit. Restricting to $\mu =\nu$, $A=B$ gives the usual symmetric homoclinic gluing bifurcation.