For a simple case of frictionless coupled oscillators shown in the figure below:
(Image: two pendula of equal length and equal masses suspended from a level ceiling and connected by a spring) (and concerning small oscillations only)
The two normal modes are $$\xi_1(t) = \frac{y_1(t) + y_2(t)}{2}$$ which is the average of the two displacements i.e. the co-ordinate of the centre of mass of the system, if you forget about the distance between the equilibrium points, and $$\xi_2(t) = \frac{y_1(t) - y_2(t)}{2}$$ which concerns the relative displacement of the pendula, and describes a motion where the centre of mass of the system does not move and the pendula swing in concert towards or away from each other.
I don't understand how you can get from this notion of relative displacement to the idea that the centre of mass of the system is not moving. I can only imagine this relative displacement varying in time and can't see how this quantity directly links to the centre of mass remaining stationary. Any pointers would be appreciated.