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Say I have a system of form $$ \frac{dy}{dt} = f(y), $$

and it is know this system has an attractor. Can I quickly for given $\varepsilon$ guess some point, such in its $\varepsilon$- neighbourhood will be present at least one point of this attractor? Quicker that integrating system itself?

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  • $\begingroup$ The attractor of a system is a very transcendental object. I doubt very much you can obtain an approximation without solving (say, numerically) said system itself. $\endgroup$ Apr 4, 2014 at 6:00
  • $\begingroup$ @LoïcTeyssier Is there any symptoms if numerical solution (or any other approximation) is nearing attractor? $\endgroup$
    – Moonwalker
    Apr 4, 2014 at 11:38
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    $\begingroup$ I saw a talk once (can't recall the author, sorry) in which the phase space was discretised, leading to successive refinement of the Lorenz attractor. It is still solving the dynamics, of course, but using an approximation of the transfer operator rather than the original ODE. Not sure if this is quicker, also will lead to different sets if there is non-transitive behaviour. I took the liberty of suggesting the "dynamical systems" tag. $\endgroup$
    – user25199
    Apr 4, 2014 at 11:52
  • $\begingroup$ @Moonwalker: not as far as I know... $\endgroup$ Apr 4, 2014 at 12:02
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    $\begingroup$ Also relevant, a recent paper: "ESTIMATING LONG-TERM BEHAVIOR OF FLOWS WITHOUT TRAJECTORY INTEGRATION: THE INFINITESIMAL GENERATOR APPROACH", SIAM J. NUMER. ANAL, 2014. $\endgroup$ Apr 4, 2014 at 13:47

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