Chaos in Time-Dependent Space-Charge Potentials

We consider a spherically symmetric, homologously breathing, space-charge-dominated beam bunch in the spirit of the particle-core model. The question we ask is: How does the time dependence influence the population of chaotic orbits? The static beam has zero chaotic orbits; the equation of particle motion is integrable up to quadrature. This is generally not true once the bunch is set into oscillation. We exemplify how the population of chaotic orbits evolves to build a halo. We also compute an integral of particle motion in this time-dependent potential, and in the process show that this computation would seem to reflect in near-real time the character of the orbital evolution, including transitions between chaos and regularity. We then introduce colored noise into the system and show how its presence modifies the dynamics.