Reduce a set of $N$ coupled differential equations to $N/2$ differential equations given a periodic boundary condition?

Question

Is it possible to reduce a set of $N$ coupled differential equations of the form $U_n''=U_{n+1}-U_{n-1}-2U_n$ to $N/2$ coupled differential equations, given the periodicity condition that $U_{N+1}$ is equivalent to $U_1$ . $''$ denotes double differential wrt a parameter $t$. May be by solving for all the odd numbered $n$s and substituting the back to the original set and reduce them to a set of just even numbered $n$s.