The Lie map generator of the dipole fringe field is derived up to the 4th order of canonical variables. We discovered significant closed orbit deviation and octupolelike potential when the bending radius ρ is small. We found that the closed orbit deviation is proportional to g 2 =ρ and the octupolelike potential effect is proportional to 1=ðgρ 2 Þ, where g is the vertical magnet gap.

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The Integrable Optics Test Accelerator (IOTA) is a novel storage ring at Fermi National Accelerator Laboratory designed (in part) to investigate the dynamics of beams in the presence of highly nonlinear transverse focusing fields that generate integrable single-particle motion with a large spread in the intrinsic betatron tunes. We describe how contemporary geometrical methods from the theory of integrable Hamiltonian systems may be used to locate all critical separatrixlike structures in the 4D transverse phase space, and to construct a complete analysis of the dynamical bifurcations of the system. Application of these techniques results in a global picture of the nominal on-energy transverse dynamics, revealing a rich diversity of accessible dynamical behavior. Similar techniques may be applied to future facilities that exploit the concept of nonlinear integrable optics.

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