Coupled Point Masses on a Circle
This Demonstration solves a system of four point masses connected with springs constrained to lie on a circular ring. Qualitatively, you can observe uniform rotation and in-phase and out-of-phase vibration, but the crossings and overlappings in the solutions are unphysical and not mentioned in textbooks.
For four masses, the equations of motion are
where is the position of the mass from its equilibrium position, is the mass of each pont, and is the spring constant.
 W. Greiner, Classical Mechanics: Systems of Particles and Hamiltonian Dynamics, 2nd ed., New York: Springer-Verlag, 2010 pp. 295–297.