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In this paper, Gian-Carlo Rota wrote:

A lot of interesting systems with constant coefficients have been discovered in the last thirty years: in control, in economics, in signal processing, even in mathematics. None of these attractive examples is presently included in introductory texts. At present all examples of matrix systems one finds in such texts are either planar or else they are artificial.

None of these attractive examples were included in this paper either.

What are some examples of what Rota is talking about?

Rota died in 1999, so "the last thirty years" can be construed accordingly, but more recent examples could be included here as well.

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    $\begingroup$ @RyanBudney : Maybe if you can make that into an answer that would be worthwhile. Believe it or not, some of us don't know about simple control systems for quadcopters. $\endgroup$
    – Michael Hardy
    Commented Nov 17, 2023 at 5:05
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    $\begingroup$ The spread of an infectious disease? I can't read Rota's mind, but I'm pretty sure "interesting" didn't mean "interesting from the mathematical point of view" but "interesting from the applications' point of view". I don't see how this is research level mathematics. Differential equations with constant coefficients aren't bleeding-edge. $\endgroup$
    – Najib Idrissi
    Commented Nov 17, 2023 at 7:27
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    $\begingroup$ The main focus of control theory is the replacement of a nonlinear system model with a linear approximation and subsequent manipulations with this linear model. $\endgroup$
    – AVK
    Commented Nov 17, 2023 at 8:29
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    $\begingroup$ @AVK: I've always thought that was called Linear Control Theory, but maybe there's varying conventions out there. To Michael, the basic idea is to think of the configuration space of a quadcopter: what the gyros say and the data from the rotors. Then you write down the ODE that tries to stabilize, i.e. make the quadcopter flat. Non-linear ODEs are often strongly controlled (conjugate) to their various linearizations, so this is a fairly general result. $\endgroup$
    – Ryan Budney
    Commented Nov 17, 2023 at 15:06
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    $\begingroup$ @GerryMyerson I wondered about the word "planar" as well. My guess is that it means a $2\times 2$ matrix. $\endgroup$
    – Timothy Chow
    Commented Nov 20, 2023 at 4:21

4 Answers 4

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There is some indication of what Rota had in mind in the book Ordinary Differential Equations by Birkhoff and Rota. I don't have a copy handy, but the preview on Amazon has this to say in the Introduction (Chapter 3 is about linear differential equations with constant coefficients):

Besides reviewing elementary material, Chapter 3 introduces the concepts of transfer function and the Nyquist diagram with their relation to Green's functions. Although widely used in communications engineering for many years, these concepts are ignored in most textbooks on differential equations.

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The first example is the harmonic oscillator. Understanding the driven damped harmonic oscillator is the foundation of electrical engineering, at least at RF frequencies.

Control theory has a number of easy systems: a thermostat that varies the energy flowing into a thermal mass that cools according to Newton's law of cooling is one. Then can ask about how particular controllers do given a step shock.

Bottom line is ask your collegues/look at the intro textbooks for those fields. There are a bunch!

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There has been some discussion of rigid-body dynamics in the comments. It's a little hard for me to imagine that there was a newly (1967+) "discovered" system of linear differential equations with constant coefficients in such a classical field; however, there was an advance in the field in the 1960s due to Thomas Kane, who found a fresh approach to analyzing rigid-body systems. Kane's method reduces computational labor in many instances, compared to classical approaches.

Rigid-body dynamics tends to be highly nonlinear. But in Kane and Levinson's book, Dynamics: Theory and Applications, there is a section on linear differential equations. In particular, there is a nontrivial example starting on page 249. Now, I'm sure one could analyze this system without using any of Kane's new ideas, so I don't think it counts as something "discovered in the last thirty years" before 1997. Nevertheless, I think it does count as a serious practical example of a system of linear differential equations with constant coefficients.

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"At present all examples of matrix systems one finds in such texts are either planar or else they are artificial." Assuming that "planar" means $2\times2$, thirty years ago I was teaching students about a trio of coupled pendulums. This leads to a $3\times3$ system, and three "eigenstates" (all three swinging together; outside one swinging in opposition with middle one stationary; outside ones swinging together while middle one swings in opposition at twice the amplitude).

I sure didn't invent this example – I must have taken it from some textbook.

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