The question is why "so many real-world systems are governed by differential equations" and "so many real-world systems involve periodic motion".

Well, stable systems are can either be stationary or oscillatory. If the world didn't contain so many stable systems, or equivalently if the laws of physics didn't allow so, then likely life would not have existed. All life is complex chemical structures, and they require stability to function. Ergo, by this anthropic argument there must be many oscillatory systems.

Differential equations aren't limited to describing stable systems, though, and there are chaotic systems that are also in some sense stable.

Ordinary differential equations can describe any system with a finite number of state variables that change continuously (as opposed to instantaneously jumping from one state to another without going through states in between) and as a function of the system's current state (as opposed to nondeterministically or under the influence of the past or future or some kind of supernatural entity).

Partial differential equations extend this to systems with infinite numbers of variables as long as the variables are organized in the form of continuous "fields" whose behavior is locally determined in a certain sense—things like the temperature that Fourier was investigating, which has an infinite number of different values along the length of an iron rod, or density, or pressure, or voltage.

It turns out that a pretty large fraction of the phenomena we experience do behave this way. It might be tempting to claim that it's obvious that the universe works this way, but that's only because you've grown up with the idea and never seriously questioned it. Consider that it isn't obvious to anyone who believes in an afterlife, or to Stephen Wolfram (who thinks continuity may be an illusion), or to anyone who bets on the lottery or believes in astrology.

But it is at least an excellent approximation that covers all phenomena that can be predicted by classical physics and most of quantum mechanics as well.

As a result, the Fourier and Laplace transforms are extremely broadly applicable, at least with respect to the physical world. In an engineering curriculum, the class that focuses most intensively on these applications is usually given the grandiose title "Signals and Systems".

That first question is a tautology. It’s like asking “Why is a screwdriver so perfect for turning screws?”

We have discovered a method (calculus) to mathematcally describe continuous functions of various sorts and within calculus there is a particular toolbox (differential and partial differential equations) we have built to mathematically describe systems that are changing by describing that change.

The fact that systems which change are well-described by the thing we have made to describe systems which change shouldn’t be at all surprising. We have been working on this since the 18th century and Euler and many other of the smartest humans ever devoted considerable effort to making it this good.

When you look at things like the chaotic behaviour of a double pendulum, you see how the real world is extremely difficult to capture precisely and as good as our system is, it still has shortcomings even in very simple cases.

> The question is why "so many real-world systems are governed by differential equations" and "so many real-world systems involve periodic motion". > > Well, stable systems are can either be stationary or oscillatory. If the world didn't contain so many stable systems, or equivalently if the laws of physics didn't allow so, then likely life would not have existed. All life is complex chemical structures, and they require stability to function. Ergo, by this anthropic argument there must be many oscillatory systems.

I would say that the it's very difficult to imagine a world that would not be governed by differential equations. So it's not just that life wouldn't exist it's that there wouldn't be anything like the laws of physics.

As a side note chaotic systems are often better analysed in the FT domain, so even in a world of chaotic systems (and there are many in our world, and I'd argue that if there wasn't life would not exist either) the FT remains a powerful tool

> Well, stable systems are can either be stationary or oscillatory.

In practice this is probably true, but I can see another possibility. The system could follow a trajectory that bounces around endlessly in some box without ever repeating or escaping the box.

Natura non facit saltus.

https://en.wikipedia.org/wiki/Natura_non_facit_saltus