When projecting a vector in Hilbert space into its (closed?) subspace, its best approximation is its Fourier series. The technique has been using in many traditional problems (heat, wave, Schrödinger) and in other low dimensional dynamical systems by finding $\lambda$ in the characteristic polynomial $det(A-\lambda I)$.

However, in general, are there any differences when applying this to any given dynamical system compared to the traditional ones? Not all systems have nice or symmetrical equations, and they may involve more variables/higher dimensions, and I think it might be stuck to find the characteristic polynomial. Is there ever such a thing, and how to solve it?

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    $\begingroup$ Fourier methods just provide one tool for approaching general differential equations, and are used to analyze many types of system. Like any other trick, they may or may not help with any particular system. Maybe the reason you've heard about just a select few is because those particular ones are such common problems and are easily treated by Fourier methods. $\endgroup$ – Mike Nov 30 '17 at 16:54
  • $\begingroup$ Is there a list of what might or might not be used with Fourier methods? $\endgroup$ – Ooker Nov 30 '17 at 20:53

The current version (v3) of the question seem to describe a particular linear approximation to the system.

If that's the case, then

  • no, there's no difference in the application of the method; and

  • it's a valid analysis, but with all limitations of local approximations.

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  • $\begingroup$ @ooker, it's hard to tell anything more specific without examples of both "traditional problems" analysis and concrete dynamical systems. $\endgroup$ – stafusa Nov 10 '17 at 16:50
  • $\begingroup$ What are the other linear approximations? The Hermit polynomials constitute spherical harmonics, and I think it's just a general case of Fourier transform $\endgroup$ – Ooker Nov 10 '17 at 17:09
  • $\begingroup$ The simplest linear approximation I can think of is a truncated Taylor series. That (in the form of $\sin{\theta}\approx \theta$) can be used to show that the harmonic oscillator is an approximation to the simple pendulum in the neighborhood of $\theta\approx 0)$. $\endgroup$ – stafusa Nov 10 '17 at 17:12
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    $\begingroup$ "the only thing?" - No, I don't think so, if only because it's such a sweeping statement. That's actually a different and more general question, this last one, which I'd rephrased as "Are all approximations describable as an expansion truncation?". The answer perhaps can be "yes" (after all, deep down perhaps we can only understand linear things), but even then, you'd probably need different descriptions in order to make different approximations, so if you want two simultaneous approximations, it doesn't necessarily work in any given single description. $\endgroup$ – stafusa Nov 10 '17 at 20:22
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    $\begingroup$ And, in general, the study of dynamical systems definitely doesn't stop at linear approximations: global dynamics, structural stability and other important measures can't be well captured by local linearizations. $\endgroup$ – stafusa Nov 10 '17 at 20:25

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