= Solution
The linear $\cos x$ mode has frequency squared $k^2-1$, so $k_c(0)=1$. To resolve the small frequency near threshold, write
$$
k=1+\varepsilon^2\kappa+O(\varepsilon^4),
\qquad
T=\varepsilon t.
$$
The leading equation
$$
-\theta_{0,xx}-\theta_0=0
$$
and the initial data give
$$
\theta_0=A(T)\cos x,\qquad A(0)=1,\qquad A'(0)=0.
$$
At $O(\varepsilon^2)$, the equation for $\theta_2$ has a forcing whose $\cos x$ component is
$$
\frac34A^3-A''-2\kappa A.
$$
The <Fredholm solvability condition> removes this <secular term> and yields the slow <amplitude equation>
$$
\boxed{
A''+2\kappa A-\frac34A^3=0
}.
$$
It has the conserved energy
$$
E=\frac12(A')^2+\kappa A^2-\frac{3}{16}A^4.
$$
The potential has maxima at
$$
A^2=\frac{8\kappa}{3}.
$$
A periodic orbit launched from $A(0)=1$, $A'(0)=0$ exists only when that turning point lies inside the two maxima, namely
$$
\kappa>\frac38.
$$
At equality the orbit is the <separatrix>; below it, the assumed real periodic oscillation is lost. Therefore
$$
\boxed{
k_c(\varepsilon)
=1+\frac38\varepsilon^2+O(\varepsilon^4)
}.
$$
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