= Asymmetric planar gluing return map
{title2=$p=\nu+Bq^\delta,\quad q=\mu+Ap^\delta$}
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>.
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