Relativistic jets, which are fast, collimated outflows that propagate over large distances, are commonly observed in various astrophysical objects, including active galactic nuclei (AGN), microquasars, and pulsars. These jets are subject to variability. We investigate scenarios with complex periodic and flicker noise variability to deepen our understanding of the physics underlying these observed variabilities. We use special relativistic simulations. We use the AMRVAC code to handle the fluid dynamics and the evolution of the electron population. In addition, we use the REPTIDE code to analyze and interpret the synthetic image from radio to X-ray. Our investigation focuses on the dynamics of variable jets. In particular, our results show for the first time that rapid jet variability can lead to the development of large quasi-steady shocks.