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

Forthe unperturbed Hamiltonian function $H=y^2/2-x^2/2+x^4/4$ satisfies the exact drift identity $\dot{H}=\epsilon (1-\alpha x^2)y^2$. The perturbation adds energy where $\alpha x^2<1$ and removes it where $\alpha x^2>1$, for positive $\epsilon$.

On a regular unperturbed periodic orbit of energy $h$, let $x_1<x_2$ be its turning points and write $y=\pm \sqrt{2h+x^2-x^4/2}$. Since $dt=dx/y$, both halves contribute the same integral:A simple zero of this energy increment selects a nearby perturbed periodic orbit by the energy balance method. The approximation is local to regular closed levels; at $h=0$, the homoclinic balance for the weakly perturbed double-well oscillator uses a convergent drift integral despite the infinite traversal time.

For $u^{\prime }=v$, $v^{\prime }=u-u^3+\epsilon (\beta -u^2)v$, let $H=v^2/2-u^2/2+u^4/4$. On an outer periodic orbit with $H>0$, leading energy balance method requires $\beta =B(H)$, whereHere $u_{\max }^2=1+\sqrt{1+4H}$. The homoclinic orbit limit is $B(0)=4/5$. Differentiating the denominator gives a period integral diverging at $H=0$, whereas the derivative of the numerator stays finite; hence $B$ decreases immediately above zero. At large $H$, rescaling $u=H^{1/4}s$ gives $B(H)\propto \sqrt{H}$. Its first nondegenerate minimum selects a saddle-node bifurcation of periodic orbits; numerical quadrature gives ${\beta }_{\mathrm{fold}}\simeq 0.75226$. The cycle on the decreasing portion is unstable and the one on the increasing portion is stable, since the net energy drift has sign $\beta -B(H)$. Restoring an unperturbed coefficient $a^2$ in the linear restoring term multiplies the threshold by $a^2$.

At energy zero, the positive homoclinic orbit of the unperturbed double-well oscillator has $y=\pm x\sqrt{1-x^2/2}$ and turning points zero and $\sqrt{2}$. The first-order drift integral from the energy balance for the weakly perturbed double-well oscillator vanishes whenBoth integrals follow by substituting $u=x^2/2$. Their ratio gives $\alpha =5/4$. The parameter derivative of the balance is $-I_2\ne 0$, so the zero is simple.

This is a leading connecting-orbit condition, rather than an exact formula for the perturbed loop at finite $\epsilon$. Under the usual nondegenerate return-map hypotheses, the Melnikov energy-balance method locates a nearby homoclinic parameter curve with $\alpha =5/4+O(\epsilon )$.

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 $1<\alpha <5/4$.