Respiratory syncytial virus (RSV) is the main cause of lower respiratory tract infections in children. Whilst highly seasonal, RSV dynamics can have either one-year (annual) or two-year (biennial) cycles. Furthermore, some countries show a 'delayed biennial' pattern, where the epidemic peak in low incidence years is delayed. We develop a compartmental model for RSV infection, driven by a seasonal forcing function, and conduct parameter space and bifurcation analyses to document parameter ranges that give rise to these different seasonal patterns. The model is sensitive to the birth rate, transmission rate, and seasonality parameters, and can replicate RSV dynamics observed in different countries. The seasonality parameter must exceed a threshold for the model to produce biennial cycles. Intermediate values of the birth rate produce the greatest delay in these biennial cycles, while the model reverts to annual cycles if the duration of immunity is too short. Finally, the existence of period doubling and period halving bifurcations suggests robust model dynamics, in agreement with the known regularity of RSV outbreaks. These findings help explain observed RSV data, such as regular biennial dynamics in Western Australia, and delayed biennial dynamics in Finland. From a public health perspective, our findings provide insight into the drivers of RSV transmission, and a foundation for exploring RSV interventions.