I would guess that you are just seeing the effects of a dynamical system with a hyperbolic fixed point.  Consider the matrix equation
$$
A'(x) = \begin{pmatrix} 0 & 1\\ \cos(x) & 0\end{pmatrix} A(x)
$$
with initial condition $A(0) = I_2$.  Because of the $2\pi$-periodicity of $\cos(x)$, this fundamental solution clearly satisfies
$$
A(x+2\pi) = A(x)A(2\pi)
$$
where 
$$
A(2\pi)\approx \begin{pmatrix} -8.065 & -8.273\\   -7.742 & -8.065\end{pmatrix},
$$
a matrix that has eigenvalues $\lambda\approx -16.068$ and $1/\lambda\approx -0.062$.  (Note that the ratio of the eigenvalues is about $\lambda^2 \approx 258$.)  The action of $A(2\pi)$ on $\mathbb{R}^2$ is hyperbolic, contracting along the $1/\lambda$-eigenspace
and expanding along the $\lambda$-eigenspace.

Now, the general solution of your equation satisfies  
$$
\begin{pmatrix}y(x)\\y'(x)\end{pmatrix}
= A(x)\begin{pmatrix}y(0)\\y'(0)\end{pmatrix},
$$
so 
$$
\begin{pmatrix}y(x+2k\pi)\\y'(x+2k\pi)\end{pmatrix}
= A(x)A(2\pi)^k\begin{pmatrix}y(0)\\y'(0)\end{pmatrix},
$$
and if the initial condition is very near (but does not lie in) the $1/\lambda$-eigenspace of $A(2\pi)$, it will take a few cycles of $2\pi$ for the very small $\lambda$-eigenvector component to grow, but once it does, it will become exponentially dominant. I'm guessing that your boundary values just happen to have hit on such an initial condition.