<p>This work explores the use of a collection of periodic and non-periodic functions for the chaotification of discrete-time systems. Chaotification is the process of composing existing chaotic maps with a new function, in order to generate new families of maps with more complex behavior. The proposed method consists of scaling a seed function, adding offset boosting, and then composing it with a boundary function. An analytical formula is provided for the Lyapunov exponent of this new family of maps. Then, several wave-type boundary functions are considered, utilizing trigonometric and polynomial terms. These are tested on a collection of seed maps, like the sine and Renyi maps. The dynamical analysis performed reveals a collection of interesting phenomena, such as robust chaos. Overall, the proposed chaotification technique will enrich the behavior of its seed map.</p>