Nonlinear Dynamical Behaviors of the Equilibrium Longitudinal   Distribution by Localized Wakes in an Electron Storage Ring
  We investigate the effects of a localized constant wake and a localized linear wake on the longitudinal beam distribution in equilibrium. We also examine the effects of the composite wake on the particle distribution when the constant wake and linear wake simultaneously exist in the ring. While moving around the parameter space, the system can show bifurcation phenomena and transition features between periodic states. We study the dynamical states in a beam, using the multi-particle tracking method in a Gaussian approximation. They show good qualitative agreement. 
S.-H. Park, H.-Ch. Kim, J.K. Ahn, and E.S. Kim