Inspired by this question:
Consider a trinary star system, in which the three stars are arranged in an xy plane and all revolving the same direction, equidistant from each other. The inward pull of gravity is balanced by the centrifugal force, establishing an orbit about a common barycentre.
Now consider adding a fourth star, displaced in the z direction directly above the barycentre of the system (along the axis of revolution). This fourth star will, as far as I can tell, be pulled downward along the axis, past the barycentre, reach a distance away from the plane of revolution equal to the distance from which it started, and begin oscillating indefinitely along the axis of revolution.
The other three stars will gain degrees of freedom as a result of this fourth star. First, they'll move up and down along the z axis as their barycentre is displaced by the gravity of the fourth. Second, they'll begin to move in and out so that their circular orbit becomes more of a sinusoidal ring, as the gravitational attraction of the fourth star will increase until it passes between the other three, then decrease as it moves away from the plane of revolution.
This star system might be best described as a modified Klemperer rosette, which can be in an ideal sense dynamically unstable - one perturbation and the system collapses. However, I’m wondering whether it’s possible for this system to be dynamically unstable in the same way that a traditional rosette is.
Can such a modified Klemperer rosette ever be dynamically unstable?
How this configuration came to be is out of the scope of the question - blame it on an incredibly powerful toddler alien who is learning to play with galaxies much as human children learn to build towers out of blocks.
Although I describe the system as "adding" a fourth star, I'm only interested in the stability of the end result (the simple harmonic oscillation of the system) and am aware that creating an additional sun in this way would destabilize the initially stable trinary star system.