Regular and Chaotic Dynamics in the Paul Trap: Fixed Points, Bifurcations, and Crystal Morphologies

The dynamics of multiple particles in a Paul-trap may often be modeled classically, leading to a system of non-linear equations describing the particles’ evolution. The non-linearity allows for the emergence of chaotic phenomena in the trap. Still, via damping, we may sometimes achieve an ordered, low-energy state of the particles, referred to as a crystal. In such a state, the particles stay nearly still, undergoing small-amplitude oscillations at the frequency of the trap’s driving force. In this Thesis, we explore, for an isolated particle species, these low-energy states. For various particle numbers, we numerically determine which regions of parameter space allow for such crystallization, and also report parameter settings where crystallization is never observed. Through observation of bifurcations and unstable fixed points, we conclude that these latter regions are areas of global chaos. We also investigate the morphologies of observed crystals, both analytically and numerically. Specifically, we report predictions of the allowable crystalline morphologies for the three, four, and five-particle systems. Generalizing methods previously used for two-particle system, we predict morphology boundaries in parameter space. By four particles, we observe the emergence of double-wells, i.e. distinct crystalline states at a fixed parameter setting.