Walkinshaw resonance is the third order difference resonance (upsilonx--2upsilonz = integer). This resonance is important in the design and operation of isochronous cyclotron and FFAG accelerators. Studying its dynamics helps us understand the mechanism of emittance growth and particle loss. This thesis studies the effect of this resonance on horizontal and vertical emittance exchange in crossing this coupling esonance. We discover the emittance scaling law and the asymptotic behaviors of this emittance exchange. Possible applications and generalizations to manipulate the beam emittance are discussed.;The thesis also studies space charge physics, which is important to high intensity and high power proton or heavy ion rings. This thesis develops a space charge tracking technique and studies physics of the beam intensity limitation. The envelope resonance can be observed by multiple particle tracking numerical simulations. We find that the beam envelope dynamics is the main issue causing the emittance growth. The quadrupole errors and the space charge perveance affect the emittance growth. In the circular accelerator, the envelope resonance stopband harmonics can be corrected by various methods. Because the envelope tunes are shifted lower by the space charge force, we find the harmonic correction with order less than twice the betatron tunes are important.
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