Application of non-linear forecasting and bispectral analysis to video observations of run-up over cuspate topography shows that these alongshore patterns in the morphology are accompanied by changes to the fundamental behaviour of the run-up timeseries. Nonlinear forecasting indicates that at beach cusp horns, the behaviour of swash flow is more predictable and global (meaning that characteristics of individual swash events are well represented by the behaviour of the timeseries as a whole). Conversely, at beach cusp bays, the behaviour of swash flow is less predictable and more local (meaning that the characteristics of individual swash events are best represented by the behaviour of a small fraction of the timeseries). Bispectral analysis indicates that there is a nonlinear transfer of energy from the incident wave frequency f to infragravity frequency ~f/2 which only occurs in the bay, suggesting that the local behaviour is caused by interactions between successive swash cycles which are magnified by channelling caused by the beach cusp geometry. The local behaviour and the bispectral signatures are not present in offshore measurements, and are not present in runup timeseries collected when the beach was planar. These results provide evidence that interactions between successive run-ups are a fundamental characteristic of beach cusp bays. Ultimately, these interactions could lead to the growth of an infragravity wave with an alongshore wavelength forced by the presence of beach cusps.