This paper continues a series reporting different aspects of the behaviour of disc galaxy simulations that support spiral instabilities. The focus in this paper is to demonstrate how linear spiral instabilities saturate and decay, and how the properties of the disc affect the limiting amplitude of the spirals. Once again, we employ idealized models that each possess a single instability that we follow until it has run its course. Remarkably, we find a tight correlation between the growth rate of the mode and its limiting amplitude, albeit from only six simulations. We show that non-linear orbit deflections near corotation cause the mode to saturate, and that the more time available in a slowly-growing mode creates the critical deflections at lower amplitude. We also find that scattering at the inner Lindblad resonance is insignificant until after the mode has saturated. Our objective in this series of papers, which we believe we have now achieved, has been to develop a convincing and well-documented account of the physical behaviour of the spiral patterns that have been observed in simulations by others, and by ourselves, for many decades. Understanding the simulations is an important step towards the greater objective, which is to find observational evidence from galaxies that could confront the identified mechanism.

The problem of how disc galaxies avoid forming bars remains unsolved. Many galaxy models having reasonable properties continue to manifest vigorous instabilities that rapidly form strong bars and no widely-accepted idea has yet been advanced to account for why this does not generally happen in most disc galaxies. The unstable mode that is responsible for bar formation is believed to be a standing wave in a cavity that reflects off the disc centre and the corotation radius, with amplification at corotation. Here we use simulations to address one further idea that may perhaps inhibit the feedback loop and therefore contribute to stability, which is to make the disc centre dynamically hot and/or to taper away mass from the inner disc, which could be masked by a bulge. Unfortunately, we find that neither strategy makes much difference to the global stability of the disc in the models we have tried. We did, however, discover one interesting wrinkle to the cavity mode: in strongly cutout discs, the wave may reflect from a radius in the disc that is clearly outside the centre, although the ingoing trailing wave still reflects into an outwardly-travelling, leading wave.