  Recently, large-scale trapped ion systems have been realized in experiments
for quantum simulation and quantum computation. They are the simplest systems
for dynamical stability and parametric resonance. In this model, the Mathieu
equation plays the most fundamental role for us to understand the stability and
instability of a single ion. In this work, we investigate the dynamics of
trapped ions with the Coulomb interaction based on the Hamiltonian equation. We
show that the many-body interaction will not influence the phase diagram for
instability. Then, the dynamics of this model in the large damping limit will
also be analytically calculated using few trapped ions. Furthermore, we find
that in the presence of modulation, synchronization dynamics can be observed,
showing an exchange of velocities between distant ions on the left side and on
the right side of the trap. These dynamics resemble to that of the exchange of
velocities in Newton's cradle for the collision of balls at the same time.
These dynamics are independent of their initial conditions and the number of
ions. As a unique feature of the interacting Mathieu equation, we hope this
behavior, which leads to a quasi-periodic solution, can be measured in current
experimental systems. Finally, we have also discussed the effect of anharmonic
trapping potential, showing the desynchronization during the collision process.
It is hopped that the dynamics in this many-body Mathieu equation with damping
may find applications in quantum simulations. This model may also find
interesting applications in dynamics systems as a pure mathematical problem,
which may be beyond the results in the Floquet theorem.
