Energy balance for the weakly perturbed double-well oscillator ($\dot{x}=y,\dot{y}=x-x^3+\epsilon (1-\alpha x^2)y$)

Forthe perturbation gives $\dot{H}=\epsilon (1-\alpha x^2)y^2$. Along an unperturbed orbit $H=H_0$ with turning points $x_1,x_2$,

On either unperturbed homoclinic loop, $H_0=0$ and the positive-side turning points are zero and $\sqrt{2}$. The first-order persistence condition iswhich gives $\alpha =5/4$.

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 $1<\alpha <5/4$.