Sometimes, it's easier to say how things change than to say how things are.
When you write down a differential equation, you're specifying constraints and information about e.g. how to model something in the world. This gives you a family of solutions, from which you can pick out any function you like, depending on details of the problem at hand.
Today, I finished the bulk of Logan's A First Course in Ordinary Differential Equations, which is easily the best ODE book I came across.
A First Course in Ordinary Differential Equations
As usual, I'll just talk about random cool things from the book.
In the summer of 2018 at a MIRI-CHAI intern workshop, I witnessed a fascinating debate: what mathematical function represents the movie time elapsed in videos like The Entire Bee Movie but every time it says bee it speeds up by 15%? That is, what mapping converts the viewer timestamp to the movie timestamp for this video?
I don't remember their conclusion, but it's simple enough to answer. Suppose counts how many times a character has said the word "bee" by timestamp in the movie. Since the viewing speed itself increases exponentially with , we have . Furthermore, since the video starts at the beginning of the movie, we have the initial condition .
This problem cannot be cleanly solved analytically (because is discontinuous and obviously lacking a clean closed form), but is expressed by a beautiful and simple differential equation.
Differential equations help us explain and model phenomena, often giving us insight into causal factors: for a trivial example, a population might grow more quickly because that population is larger.
Equilibria and stability theory
This material gave me a great conceptual framework for thinking about stability. Here are some good handles:
Let's think about rocks and hills. Unstable equilibria have the rock rolling away forever lost, no matter how lightly the rock is nudged, while locally stable equilibria have some level of tolerance within which they'll settle back down. For a globally stable equilibrium, no matter how hard the perturbation, the rock comes rolling back down the parabola.
A familiar example is a playground swing, which acts as a pendulum. Pushing a person in a swing in time with the natural interval of the swing (its resonant frequency) makes the swing go higher and higher (maximum amplitude), while attempts to push the swing at a faster or slower tempo produce smaller arcs. This is because the energy the swing absorbs is maximized when the pushes match the swing's natural oscillations. ~ Wikipedia
And that's also how the Tacoma bridge collapsed in 1940. The second-order differential equations underlying this allow us to solve for the forcing function which could induce catastrophic resonance.
Also note that there is only at most one resonant frequency of any given system, because even lower octaves of the natural frequency would provide destructive interference a good amount of the time.
- This book gave me great chance to review my calculus, from integration by parts to the deeper meaning of Taylor's theorem: that for many functions, you can recover all of the global information from the local information, in the form of derivatives. I don't fully understand why this doesn't work for some functions which are infinitely differentiable (like ), but apparently this becomes clearer after some complex analysis.
- Bifurcation diagrams allow us to model the behavior, birth, and destruction of equilibria as we vary parameters in the differential equation. I'm looking forward to learning more about bifurcation theory. In this video, Veratasium highlights stunning patterns behind the bifurcation diagrams of single-humped functions.
I supplemented my understanding with the first two chapters of Strogatz's Nonlinear Dynamics And Chaos. I might come back for more of the latter at a later date; I'm feeling like moving on and I think it's important to follow that feeling.