There are lots of examples of oscillatory phenomena in nature the description of which boils down to simple harmonic behavior, i.e. to Cosine/Sine/Complex Exp. This answer explains that we use sines and cosines because of their convenience and ease of mathematical description of phenomena. For example, complex exponentials are eigenfunctions of the differentiation operator, so linear differential equations turn into algebraic ones. In other words, we usually describe the complex vibration of a drum as a sum of sinusoidal modes, because it is convenient for us. But these drum vibrations could also be described as the sum of triangular or rectangular modes or any other modes (as long as the corresponding functions form a complete basis).

But my question is different: are there such physical systems in which harmonic waves, described by sines and cosines, cannot exist? Let me rephrase it. For example, we consider an oscillating system. It can be linear, and then its dynamics are described by a linear differential equation. It can also be nonlinear, then we can linearize the system and consider the stability of the stationary state. In any case, we arrive at eigenvalues and eigenfunctions. Is it possible that sines and cosines are not proper functions for this system? Maybe you know at least a couple of examples of such systems? Occasionally, in the literature on mechanics, I come across that there is a class of phenomena, the description of which is not limited to the study of the properties of individual harmonic waves. But no examples are given...