
Interview with Dr Erika Camacho on the Hartman-Grobman theorem [audio] - extarial
https://blogs.scientificamerican.com/roots-of-unity/the-most-addictive-theorem-in-applied-mathematics/
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heisenbit
Some people are stuck in black and white thinking. They struggle to visualize
the gradual transition that often happens in real life.

Some people are stuck in seeing everything gradual. They tend to discount
systems flipping and long tail events.

Then there a few who see mostly nonlinear systems and neglect the spoils that
lay in linear analysis of a localized space.

And on microscopic level there is the weirdness of quantum effects.

Knowing into which toolbox to reach is not always clear.

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andrepd
I've no idea what you mean by this. It's confusing to me.

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super_mario
TLDR: The "most addictive theorem" is the Hartman–Grobman theorem which states
that under some very specific and technical conditions non-linear system near
equilibrium can be described with a linear system.

More at Wikipedia:
[https://en.wikipedia.org/wiki/Hartman–Grobman_theorem](https://en.wikipedia.org/wiki/Hartman–Grobman_theorem)

~~~
jesuslop
Rings a bell on Progogine parlance about Onsager relations.

~~~
jesuslop
Prigogine, I meant

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neonate
Transcript at [https://kpknudson.com/my-favorite-
theorem/2018/9/12/episode-...](https://kpknudson.com/my-favorite-
theorem/2018/9/12/episode-26-erika-camacho).

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golemotron
Does anyone have a TL;DL?

~~~
mike_n
Imagine you're looking at a ball rolling around on a complicated curved
surface defined by differential (non-linear) equations, which can be a tricky
system to analyze.

If the ball is near some sort of a saddle equilibrium point, the theorem says
that you can simplify things by flattening out a small patch of the surface if
you are (very) near that point.

This is a lot easier to analyze using simple linear algebra tools, and still
gives good results for predicting what happens next.

~~~
igivanov
Generally it's a pretty basic concept in analyzing any system, 1st step is to
use a constant, if it doesn't fit then a linear model, etc. Can use different
approximations depending on the circumstance, e.g. you can approximate sin(x)
with x around 0...

Is there something special to it in this particular case?

~~~
chestervonwinch
Hartman-Grobman isn't so much about approximation error by choosing different
simplifying models. It's about being able to do some analysis on the linear
approximation with the results carrying over to the original non-linear
system. In particular, it lets you categorize stationary points of a non-
linear dynamical system, using only the linear approximation at the fixed
points, which is awesome since analyzing linear systems is super easy.

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enriquto
> In mathematics, nonlinear basically means “hard to analyze;”

my god how low Scientific American has fallen

~~~
simulate
Not sure what your issue with "hard to analyze" is. When compared to linear
systems, nonlinear dynamical systems _are_ hard to analyze.

> Nonlinear dynamical systems, describing changes in variables over time, may
> appear chaotic, unpredictable, or counterintuitive, contrasting with much
> simpler linear systems.

[https://en.m.wikipedia.org/wiki/Nonlinear_system](https://en.m.wikipedia.org/wiki/Nonlinear_system)

~~~
theoh
"Hard to analyze" is a property nonlinear systems have. It's not the meaning
of the term.

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skh
That is not the meaning of the term. However, if you are to describe what the
big deal is about finding ways to analyze nonlinear systems to lay people then
what better description is there?

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theoh
First line of the Wikipedia page is fine: "In mathematics and science, a
nonlinear system is a system in which the change of the output is not
proportional to the change of the input."

Scientific American's typical reader surely understands a) the concept of a
function b) the concept of proportionality.

It's fine to say that nonlinear systems are difficult to analyze, or have a
reputation for being difficult to analyze. But it's bad practice for a science
communicator even to run the risk of giving people the impression that the
difficulty is part of the meaning (definition) of the word.

~~~
alew1
Did the article change? The rest of the “hard to analyze” sentence you quoted
above is “linear systems respond proportionally to changes in variables,
whereas nonlinear systems have more complicated relationships.“

