ODE to Joy: Insights from ‘A First Course in Ordinary Differential Equations’

Foreword

Some­times, it’s eas­ier to say how things change than to say how things are.

From 3Blue1Brown: Differ­en­tial Equations

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.

Bee Movie

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.

Gears-level mod­els?

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.

Resonance

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.

Ran­dom notes

  • 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.

Forwards

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.