Systems of Differential Equations · Superchallenge problem

Two coupled masses obey $x''=-\tfrac{13}{2}x+\tfrac52y,\ y''=\tfrac52x-\tfrac{13}{2}y$. Find the smallest $T>0$ such that every solution satisfies $\big(x,y,x',y'\big)(T)=\big(x,y,x',y'\big)(0)$.