At the saddle equilibrium , put . The two eigenvalues and saddle index are
On the nearby saddle-loop curve, . Therefore , with the more explicit expansion
The positive saddle quantity makes the loop repelling. In a transverse section its leading return map has the form ; its derivative grows without bound as . This is the mechanism by which a positive saddle quantity makes a nearby saddle loop repelling. It agrees with the repelling periodic orbit coming from the subcritical Hopf bifurcation, rather than predicting an attracting cycle at the collision.