Beam dynamics problems relevant to the understanding of the coherent beam instabilities and the nonlinearity in the Cooler Ring will be investigated. These research topics are indispensable to the effort of increasing luminosity at the IUCP Cooler Ring and are also important to modern high brightness storage rings. The proposed research is also important in training the accelerator physics graduate and undergraduate students. The task A studies "Impedances and Instabilities measurement of the Cooler". Topics of interest are coherent instabilities and preservation of small emittance beam in high energy (and high brightness) accelerators. Using the measured beam transfer function, the impedance can be deduced. Model simulation and analytic evaluation of the Vlasov equation will be compared with experimental data. These experiments will help to understand better the current limitation of the Cooler Ring. It will also provide information for the design of damper system. Through this series of experiments, we will be able to understand the dynamics of coherent instability and multidimensional Landau damping on the coherent growth. These experiment can also be important to Plasma physics. The task B studies "Phase and Amplitude Modulations in Parametric Resonances". The experiment is to measure beam responses to the phase or amplitude modulations in the rf system. These response functions are solutions of parametric resonant equations. These experiments are important to understand the diffusion process in the longitudinal phase space. Using the phase feedback loop, highly damped parametric resonant system can be studied as well. Phenomena such as chaos, strange attractors, and bifurcations can be measured. With these experimental tools, we can set up an excellent laboratory for nonlinear dynamics.
