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Interview with Dr Erika Camacho on the Hartman-Grobman theorem [audio]
- golemotron 8y agoDoes anyone have a TL;DL?
- mike_n 8y agoImagine 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 8y agoGenerally 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 8y agoHartman-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.
- salty_biscuits 8y agoMy take on this theorem is that it is most interesting to know when the topological equivalence doesn't hold, i.e. linear approx of equilibrium point can be poor when there is the possibility of a limit cycle oscillation.
- enriquto 8y ago> In mathematics, nonlinear basically means “hard to analyze;” my god how low Scientific American has fallen
- Kenji 8y agoWell I've seen worse rules of thumb.
- raverbashing 8y agoI think it's a bit of a tongue in cheek statement tbh
- simulate 8y agoNot 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 8y ago"Hard to analyze" is a property nonlinear systems have. It's not the meaning of the term.
- skh 8y agoThat 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?
- theoh 8y agoFirst 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.
- super_mario 8y agoTLDR: 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
- heisenbit 8y agoSome 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.
- neonate 8y agoTranscript at https://kpknudson.com/my-favorite-theorem/2018/9/12/episode-26-erika-camacho https://kpknudson.com/my-favorite-theorem/2018/9/12/episode-....