Circular accelerators, such as the ones in the Fermilab Accelerator Complex, are indispensable tools driving scientific discovery by propelling particles to high energies and high intensities. In the context of the Proton Improvement Plan II (PIP-II) at Fermilab, the Recycler Ring (RR) is confronted with the challenge of high-intensity operation. The space charge tune shifts at these intensities necessitate the mitigation of third-order resonances. By measuring the Resonance Driving Terms (RDTs) for each resonance line, their strengths and damaging effects can be quantified. The study is focused on the utilization of dedicated normal and skew sextupoles for the compensation of these resonance lines. By employing the response matrix method to measure the Resonance Driving Terms (RDTs) relative to the currents in these sextupoles, a methodical approach to simultaneously compensate individual and multiple third-order resonances has been developed and assessed. The assessment and verification incorporate mapping out beam losses through dynamic and static tune scans, along with the utilization of the Ion Profile Monitor (IPM) system for beam size measurements. Furthermore, optimization algorithms have been explored and incorporated into the resonance compensation scheme. Parallel, to the work done at Fermilab, the study extends its scope by performing related experiments at the CERN Proton Synchrotron Booster (PSB). For these experiments, multiple resonance lines in the tune diagram were compensated by optimizing the currents in the compensation elements with the aid of advanced optimization algorithms. High-intensity operation of the Recycler Ring involves understanding the interplay between the space charge tune spread, resonance lines, and specialized subsystems such as transverse dampers. The following study also delves into experiments that try to discern each of these phenomena, while highlighting challenges still ahead. Overall, these studies indicate the feasibility of compensating for multiple third-order resonances, contributing to the objectives of PIP-II for the reliable delivery of a 1.2 MW proton beam to experimental facilities.
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