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A nearly Hamiltonian system is a weak perturbation of a Hamiltonian system. Along its paths, $\dot{H}=ϵ\nabla H\cdot R$. Averaging this drift over an unperturbed periodic orbit is the energy balance method; a simple zero of the averaged drift selects a persistent nearby periodic orbit for sufficiently small $ϵ$. A multiple zero requires higher-order analysis.

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 $a^2$.

Set $H=({\dot{x}}^2+x^2)/2$. ThenOn an unperturbed circle write $x=r\cos t$, $\dot{x}=-r\sin t$. The supplied integrals giveEquivalently, the leading averaged amplitude equation isWith $s=r^2$, its nonzero stationary amplitudes solve $\beta s^2-2\alpha s+8=0$. If ${\alpha }^2>8\beta$, both roots are positive and simple:The averaged drift is positive below $r_{-}$, negative between $r_{-}$ and $r_{+}$, and positive above $r_{+}$. Thus the inner orbit is attracting and the outer orbit repelling, with leading displacement amplitudes $r_{-}$ and $r_{+}$ and period $2\pi +O(ϵ)$. The simple zeros of the Poincaré return map persist for sufficiently small $ϵ$.

If ${\alpha }^2<8\beta$, the leading quadratic is strictly positive for all $s\ge 0$; the energy increases on every nonzero unperturbed orbit, so the energy balance method yields no periodic orbits in any fixed compact range of nonzero amplitudes for sufficiently small $ϵ$. This perturbative calculation does not itself establish a uniform global exclusion at amplitudes growing as $ϵ$ tends to zero. At equality, the leading drift has a double zero at $r^2=\alpha /\beta$. It is positive on both sides, with a semistable candidate at that amplitude. First-order averaging alone cannot decide whether the exact system has zero, one, or two nearby periodic orbits at equality: higher-order drift determines the unfolding of this degenerate zero. The original PDF has ${\alpha }^2=8\beta$; the extra exponent on $\beta$ in the converted TeX is an OCR error.

For a nearly Hamiltonian system, the unperturbed Hamiltonian is conserved and its level curves typically contain a family of periodic orbits. A weak perturbation changes the energy slowly. The energy balance method integrates $\dot{H}$ over one unperturbed period to obtain the leading energy change $\Delta H(H)$. A zero of this change selects a possible persistent periodic orbit. A simple zero persists for sufficiently small perturbation by a Poincaré return map and the implicit function theorem. A negative slope of the averaged energy drift gives attraction; a positive slope gives repulsion. A multiple zero requires higher-order analysis, and a zero-free leading drift rules out periodic motion in the corresponding perturbative energy range.

Use the nearly conserved oscillator energy $E=({\dot{x}}^2+x^2)/2$. The differential equation givesFor a leading-order orbit $x=A\cos \theta$, $\dot{x}=-A\sin \theta$, the cycle averages are $⟨{\dot{x}}^4⟩=3A^4/8$, $⟨x^3\dot{x}⟩=0$, $⟨x^4{\dot{x}}^2⟩=A^6/16$, and $⟨{\dot{x}}^2⟩=A^2/2$. The energy balance method therefore givesThe two nonzero simple roots give the leading limit cycles $x\simeq \cos (t+{\theta }_0)$ and $x\simeq \sqrt{2}\cos (t+{\theta }_0)$. Below amplitude $1$ the energy increases, between $1$ and $\sqrt{2}$ it decreases, and above $\sqrt{2}$ it increases. Hence the amplitude-$1$ cycle is stable and the amplitude-$\sqrt{2}$ cycle unstable for positive small $\epsilon$.

The linearized averaged radial exponents are $-\epsilon /2$ and $\epsilon$, respectively, consistent with nontrivial Floquet multipliers $e^{-\pi \epsilon +O({\epsilon }^2)}$ and $e^{2\pi \epsilon +O({\epsilon }^2)}$. The nonzero derivatives at these averaged zeros also justify persistence of nearby cycles for sufficiently small $\epsilon$.

For the unperturbed Hamiltonian system, choose a conserved energy $H(x,y)$. Along the perturbed system,The energy balance method integrates this expression around each unperturbed periodic orbit. A zero of the resulting $O(\epsilon )$ energy change is a necessary leading-order condition for a nearby periodic orbit. A simple zero at which the drift changes from positive inside to negative outside predicts a stable orbit; the opposite change predicts an unstable one.