= Solution
Use the <Poincaré-Lindstedt method>. Introduce $\theta=\Omega t$ and $x=x_0(\theta)+\epsilon x_1(\theta)+\cdots$. The leading equation gives $\omega_0=3$ and $x_0=A_0\cos\theta$, after fixing the arbitrary <phase>. At order $\epsilon$,
$$
9(x_1''+x_1)=6\omega_1A_0\cos\theta+3A_0(A_0^2-1)\sin\theta+3A_0^3\sin3\theta.
$$
Here $4\cos^2\theta\sin\theta=\sin\theta+\sin3\theta$. Periodicity requires the two resonant coefficients to vanish. With $A_0>0$, this gives
$$
\boxed{A_0=1,\qquad\omega_0=3,\qquad\omega_1=0.}
$$
Therefore the leading <limit cycle> is $x(t)=\cos(3t)+O(\epsilon)$, with frequency $3+O(\epsilon^2)$. For example the nonresonant first correction contains $-\epsilon\sin(9t)/24$; its homogeneous <sine>/<cosine> pieces depend on <phase> and <amplitude> conventions. The absence of a first-order frequency shift is a solvability condition, not an assumption.
Back to article page
