Sufficient conditions for the existence of periodic motion of linear time-invariant systems when represented in phase variable form are obtained through Liapunov's second method. A method of constructing Liapunov functions to obtain conditions for non-asymptotic, non-periodic stability of these systems is suggested. Matrix analogues of these conditions serve to determine the orbital motion and non-asymptotic non-periodic stability of coupled systems whose matrix coefficients are simultaneously diagonalizable.