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In every textbook on chaos, there are a lot of numerical simulations. A typical example is the Poincare section.

But why is numerical simulation still meaningful if the system is very sensitive to numerical errors?

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The weather is a chaotic system, but we can still predict it reasonably well for tomorrow and the day after tomorrow. However, it becomes very difficult to predict what type of weather we will have in 2 months from now. – Hunter Jun 21 '14 at 15:28

3 Answers 3

Numerical simulations are not always meaningful, as chaos theory belongs to the large subject of dynamical systems theory. Although the definitions differ, chaos generally occurs in three contexts:

  1. Sensitive dependence on initial conditions (SDIC).
  2. The set is topologically transitive.
  3. Periodic points are dense in the set.

Think of two particles having some trajectory. Their motion will be chaotic if with some small perturbation in their position do the trajectories diverge. Your question specifically is about numerical errors, which is related to criteria 1 above. Namely, SDIC implies that a function $f$ has SDIC if there exists some $\delta > 0$, such that $|f^{n}(x) - f^{n}(y)| > \delta$. It turns out that $\delta$ actually is not so small that most good numerical solvers can not handle the numerical simulations. Since chaos theory occurs in most dynamical systems which are typically coupled autonomous ordinary differential equations, there is a wide array of solvers that are able to handle this problem quite well, such as the Runge-Kutta solvers which have a very small numerical error associated with them.

As I said above, which is related to the second part of your question, numerical simulations are not always necessary, most textbooks use them for demonstration purposes to get students interested in dynamical systems. For example, you can generally show a dynamical system to exhibit chaotic motion if the underlying geometry of the geodesics has negative Ricci curvature. This is for Hamiltonian systems only, but, one can always extend a non-Hamiltonian system to a Hamiltonian system by extending the phase space. One can also obtain a whole wealth of information about a system and whether it exhibits chaos by calculating the alpha and omega-limit sets, finding the future and past attractors via Lyapunov, Chetaev, functions, and applying the LaSalle invariance principle. If you would like to read up more on this, the book by Hirsch, Smale, and Devaney is very good.

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Three different points of views on essentially the same thing:

  • Chaotic systems are not only sensitive to numerical errors, but also to any other small perturbations, such as dynamical noise, which may simulate real conditions.

  • Though tiny perturbations affect the detailled, microscopic future of a system, its qualitative dynamics is unaffected. And the latter is what we want to investigate, if we simulate a chaotic system.

  • The butterfly effect only is a problem, if you want to precisely predict the future of a system. But this is something, which we cannot do anyway in reality due to, well, reality being noisy.

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I would connect the second point with the property called ergodicity, exhibited by some (or many) chaotic systems. If you are interested in some qualitative features and use an energy conserving simulator, it is not a big issue if your numerical simulation diverges from the exact solution, as long as you are on the same energy shell in phase space since the exact solution would cover the whole energy shell anyway. – Tarek Jun 21 '14 at 15:37
The perturbations do affect the qualitative dynamics of a system, this is the point of studying chaos dynamics. Depending on our knowledge of the system, we can find the effect the perturbations have on various properties of the system so that this effect is diminished. For example, numerical integration of a 2-body system can cause the one body to be expelled from the gravity field due to accumulation of errors that make the energy positive. – auxsvr Jun 21 '14 at 17:52
@auxsvr: The majority of studies of chaotic systems I am aware of do not study the effect on tiny perturbations (on the scale of numerical errors) on the qualitative dynamics (though perturbations are used to estimate properties such as Lyapunov exponents, but that’s a different story). Also, the 2-body system is not chaotic. – Wrzlprmft Jun 22 '14 at 8:16

It is a valid question to ask whether the computer simulation of a dynamical system is representative of the dynamical behavior of the real system or merely the artifact of roundoff errors caused due to the necessarily finite precision of a real computer.

There is a crucial result regarding this situation called the shadowing theorem [1]. It states that

Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near ("shadows") the numerically computed one.

So when I iterate a chaotic dynamical system starting from an initial condition P, the trajectory that the computer spits out may not be representative of the real position of the dynamical system due to the roundoff errors. However, what there will exist an initial condition Q, such that the real trajectory starting from Q will stay close to my computer generated trajectory from P.

This tells me that if my computer simulation shows me a fractal structure of curves, this structure really is shown by the real dynamical system in that there exist trajectories that shadow the trajectories shown by my computer.

[1] (1993) Ott, E. Chaos in Dynamical Systems, Cambridge University Press, pages 18-19.

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