Progress in the development of Interaction Regions (IR) for advanced future colliders, in particular the Muon Collider and the International Linear Collider, require a systematical high-order dynamics study. The effects from high-order beam dynamics become more and more pronounced as the spot size at the Interaction Point decreases to very small values, 14 cm and 4 cm for the 50 on 50 GeV Muon Collider and Higgs Factory, respectively, and even less for the International Linear Collider. Simultaneous control of geometric and chromatic aberrations is critical to the success of future machines and can only be achieved through the deliberate addition of nonlinear fields in the Interaction Region itself. This work studies both the correction schemes and the unavoidable impact of high-order correctors - sextupoles, octupoles and even duodecapoles - located in the Interaction Region close to the low-beta quadrupoles or focusing elements. This study proposes and systematically addresses the evolution of aberrations for different systems of nonlinear correctors and optimizes performance of advanced IRs.
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