• The new idea in the space charge dominated beam problem is to express the KV envelope equation in Hamiltonian formalism. Using the KV Hamiltonian, the mis-matched beam envelope function can be studied. Expanding the KV Hamiltonian in action-angle variables, parametric resonances appear as perturbation to Hamiltonian tori. These parametric resonances form families of bifurcation trees (see S.Y. Lee and A. Riabko, Phys. Rev. E 51, 1609 (1995)).

• For particle motion in the space charge dominated beams, the coupled equation between the envelope degree freedom and the particle phase space coordinates can create chaos. When resonances overlap with each other, global chaos occurs. The global chaos can result in beam halo observed in the low energy beam transport channels. (See A. Riabko et al., Phys. Rev. E51 3529 (1995)).

• More recently, we are working on the envelope dynamics of the FOFO focusing channel. We studied the Mathieu instability and effect of higher order parametric resonances. The following figure shows the condition below, in the region of, and above the mathieu envelope instability. (A. Riabko et al., paper submitted to the Nucl. Instr. and Methods, 1995) .