= Solution
Near the first <eigenvalue>, load detuning supplies a term $\epsilon\lambda_1\theta$, while the first nonlinearity in $\sin\theta=\theta-\theta^3/6+\cdots$ is cubic. Balancing them gives $\theta^2=O(\epsilon)$, so $\boxed{\theta=O(\sqrt\epsilon)}$.
Put $\theta=\sqrt\epsilon[A_0\sin\pi\xi+\epsilon\Theta_1+\cdots]$. At order $\epsilon^{3/2}$,
$$
(\partial_\xi^2+\pi^2)\Theta_1=-\pi^2\lambda_1A_0\sin\pi\xi+\frac{\pi^2A_0^3}{6}\sin^3\pi\xi.
$$
The operator has <kernel> $\sin\pi\xi$, so Fredholm solvability requires its forcing to be <orthogonal> to that function. Since $\int\sin^2\pi\xi=1/2$ and $\int\sin^4\pi\xi=3/8$,
$$
-\lambda_1A_0/2+A_0^3/16=0,\qquad\boxed{A_0(A_0^2-8\lambda_1)=0.}
$$
When this holds the nonresonant forcing is $-\pi^2A_0^3\sin3\pi\xi/24$; for instance $\Theta_1=A_0^3\sin3\pi\xi/192$ is a solution after absorbing its arbitrary first-mode component into the <amplitude> convention.
The branches are $\boxed{A_0=0\text{ for all }\lambda_1,\qquad A_0=\pm\sqrt{8\lambda_1}\text{ for }\lambda_1\geq0}$. This is a <supercritical pitchfork bifurcation>. The leading reduced energy is proportional to $A_0^4-16\lambda_1A_0^2$: the nonzero branches are minima for $\lambda_1>0$, while zero loses minimality on that side. At the origin the quartic term still stabilizes the critical state.
\Image[/past-exam-of-the-elasticity-and-plasticity-course-of-the-university-of-oxford/2008-strut-pitchfork.png]
{title=Supercritical strut response: stable nonzero branches emerge above the critical load; dashed zero branch is unstable}
Back to article page
