Monday 25 July 2016

differential equations - How are Fourier transforms of any dynamical system different to traditional ones?


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?



Answer



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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