= Solution
The equilibrium equations give $y=0$ and $x-x^3=0$, so the fixed points are
$$
\boxed{(0,0),\qquad(1,0),\qquad(-1,0).}
$$
The Jacobian is
$$
J(x,0)=
\begin{pmatrix}
0&1\\
1-3x^2&\varepsilon(1-\alpha x^2)
\end{pmatrix}.
$$
At the origin its determinant is $-1$, so the origin is a <saddle equilibrium>. At either of the other points the determinant is two and the trace is $\varepsilon(1-\alpha)$. For small positive $\varepsilon$, they are unstable foci when $\alpha<1$ and stable foci when $\alpha>1$; at $\alpha=1$ the linearization is nonhyperbolic. When $\varepsilon=0$, each is a <center equilibrium>.
For $\varepsilon=0$, the system is <Hamiltonian> with
$$
\boxed{H(x,y)=\frac12y^2-\frac12x^2+\frac14x^4,}
$$
because $\dot x=H_y$ and $\dot y=-H_x$. Its <conservative planar phase portrait> has two wells centered at $(\pm1,0)$, a saddle at the origin, periodic level curves around each center for $-1/4<H<0$, two homoclinic loops on $H=0$, and outer periodic curves surrounding both wells for $H>0$.
For positive small $\varepsilon$,
$$
\dot H=H_x\dot x+H_y\dot y
=\varepsilon(1-\alpha x^2)y^2.
$$
On the unperturbed level $H=H_0$,
$$
y^2=2H_0+x^2-\frac12x^4.
$$
The upper and lower halves contribute equally because $dt=dx/y$. Hence the <energy balance for the weakly perturbed double-well oscillator> is
$$
\Delta H
=2\varepsilon\int_{x_1}^{x_2}
(1-\alpha x^2)
\sqrt{2H_0+x^2-\frac12x^4}dx+O(\varepsilon^2),
$$
so one may take
$$
\boxed{F(x;\alpha,H_0)=
2(1-\alpha x^2)
\sqrt{2H_0+x^2-\frac12x^4}.}
$$
For the right homoclinic loop, $H_0=0$, $x_1=0$, and $x_2=\sqrt2$. Its persistence requires
$$
0=\int_0^{\sqrt2}(1-\alpha x^2)x
\sqrt{1-x^2/2}dx.
$$
The substitution $u=1-x^2/2$ gives
$$
\int_0^{\sqrt2}x\sqrt{1-x^2/2}dx=\frac23,
\qquad
\int_0^{\sqrt2}x^3\sqrt{1-x^2/2}dx=\frac8{15}.
$$
Thus the <homoclinic balance for the weakly perturbed double-well oscillator> occurs at
$$
\boxed{\alpha=\frac{2/3}{8/15}=\frac54.}
$$
The left loop gives the same condition by symmetry.
For an orbit surrounding one center, the corresponding balance ratio tends to $\alpha=1$ as the orbit shrinks to the center and to $\alpha=5/4$ as it approaches the homoclinic loop. The <single-well periodic orbit range for the weakly perturbed double-well oscillator> is therefore
$$
\boxed{1<\alpha<\frac54.}
$$
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