The radiation belts contain high energy charged particles that are constantly lost and accelerated through a multitude of external factors, such as interactions between waves and particles. Most current radiation belt models are based on a quasilinear theory framework. In some cases, this quasilinear simplification cannot account for the dynamics of the belts, and it is helpful to turn to analysing nonlinear effects. Current theories assume wave-particle interaction occur only at perfect resonant frequencies. This does not work when modelling particles moving at an angle at or near to 90 from the magnetic field line. This PhD involves applying ideas of expanding the range of frequencies at which we model waves and particles interacting. This means utilising a concept known as resonance broadening to expand quasilinear theories and running simulations to assist development of radiation belt models and provide a better physical understanding of the dynamics of plasmas.