Single-well periodic orbit range for the weakly perturbed double-well oscillator

For the periodic energy levels $-1/4<H_0<0$ enclosing one well, the energy-balance value of $\alpha$ ranges from one at the center to $5/4$ at the homoclinic loop. Thus for small positive $\epsilon$, one expects one symmetric pair of single-well periodic orbits when $1<\alpha <5/4$.