Viruses evolve by periods of relative stasis interleaved with sudden, rapid series of mutation fixations, known as evolutionary bursts. These bursts can be triggered by external factors, such as environmental changes, antiviral therapies, or spill-overs from reservoirs into novel host species. However, it has also been suggested that bursts may result from the intrinsic evolutionary dynamics of viruses. Indeed, bursts could be caused by fitness valley crossing, or a neutral exploration of a fitness plateau until an escape mutant is found. In order to investigate the importance of these intrinsic causes of evolutionary bursts, we used a simulation software package to perform massive evolution experiments of viral-like genomes. We tested two conditions: (i) after an external change and (ii) in a constant environment, with the latter condition guaranteeing the absence of an external triggering factor. As expected, an external change was almost systematically followed by an evolutionary burst. However, we also observed bursts in the constant environment as well, albeit much less frequently. We analyzed how many of these bursts are triggered by deleterious, quasi-neutral, or beneficial mutations and show that, while bursts can occasionally be triggered by valley crossing or traveling along neutral ridges, many of them were triggered by chromosomal rearrangements and, in particular, segmental duplications. Our results suggest that combinatorial differences between the different mutation types lead to punctuated evolutionary dynamics, with long periods of stasis occasionally interrupted by short periods of rapid evolution, akin to what is observed in virus evolution.