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Can anyone suggest a good, concrete example of using "chaos theory" to solve an easily understood engineering problem?

I'm wondering if there is a an answer of the following sort:

"We have a high level objective to design a system that does XYZ. To achieve this objective, we propose the following design D. If we look at the low level physics of our design, we see that the dynamics are nonlinear and exhibit "chaotic" behavior, but we can categorize the chaotic behavior as of being type ABC. Because we are able to understand that the chaotic dynamics are of the specific type ABC, although we can't make precise statements about every aspect of the system, we are still able to make the following "high-level" claims. And using those claims, we have designed a system in which,although certains parts are behaving "chaotically," the system still performs our desired objective XYZ very effectively."

I'm not looking for answers of either the following sort:

"Looking at the underlying physics we see that the dynamics are chaotic, but we can also see that if we introduce a mechanism EFG into our design, we can see that it will "dampen" the chaotic behavior and then leave us in a place where we can find a good solution."

or

"Looking at the underlying physics, we can see that certain aspects might be chaotic, but it turns out for the following reasons R1, R2, etc that underlying chaos has no bearing on our high level objective and we can create a solution in which we do not need to worry about the chaotic aspects."


An ideal answer might be something like "even though it's clear that this wing design is going to create a lot of turbulence, we can see that the turbulence will have a certain "structure", and thinking about this "structure" just a little bit, we can see that it's going actually work in our favor and greatly increase fuel efficiency." And course the problem is that it seems so implausible that turbulent flow ever works out this way -- rather it is something that is meant to be stamped away at all costs, but I imagine there must be some other example out there.

I find most texts on nonlinear dynamics/chaos very reasonable in their mathematical development of why chaos occurs, and how it has certain structure, but while I find myself able to see ways of avoiding chaos or even ignoring it, I don't have a good idea of how I could use my knowledge of it's structure directly to my advantage.

thanks!

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How is that setting called when you put a ring on a sphere to make boundary layer turbulent so it would separate later to minimize drag? –  Yrogirg Aug 22 '12 at 4:32
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@Yrogirg: That's also the only example I could think of, making a fluid turbulent on purpose. You should make it an answer. The issue is that this type of chaos is not particularly well understood from anything that could be called "Chaos theory", just phenomenologically, and I don't think there are any examples using bifurcations or attractor shapes explicitly. –  Ron Maimon Aug 22 '12 at 6:16
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2 Answers 2

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The chaos theory enables 2 broad families of engineering applications.

  • First bases on the fact that low frequence periodic unstable orbits are embedded in each chaotic attractor. In other words there are very simple and periodic dynamical modes inside the complex chaotic behaviour. The idea is then to perturb the system so that it moves towards periodic orbits of desired frequency. One of the advantages is the extreme sensibility of the system so that with very small perturbations can be obtained very large effects. Consequence is that the system may be moved to the desired state very fast and with low energy cost. Among existing applications is cryptography.

  • Second bases on attraction bassins of multifractals. Here the idea is to use perturbations to guide the system towards the desired attractor when several exist. Both families belong to the broad concept of "chaos control".

A very comprehensive and rather technical description of methods, applications and justification may be found here : http://hildalarrondo.net/wp-content/uploads/2010/05/PhysRepBoccaletti2000.pdf

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This paper is very nice -- particularly Section 6 which covers a few empirical studies. –  lilinjn Jun 28 '13 at 15:03
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A stepping mortor or magnetic field lines in a Tokamak when two magnetic islands overlap are simple examples of chaotic behaviour (really different from turbulent) that can be easily controlled.

http://www.epj.org/_pdf/HP_EPJB_slowly_rocking.pdf

http://www-student.elec.qmul.ac.uk/people/josh/documents/ReissAlinSandlerRobert-ICIT2002.pdf

Lyshevski S., «Motion control of electromechanical servo-device with permanent-magnet stepper motors», Mechatronics vol.7 n°6 1997, pp 521-536

Pera M.C., Robert B., Goeldel C., « Nonlinear dynamics in electromechanical systems-application to a hybrid stepping motor » Electromotion, 7, 31-42, 2000

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thanks, the two links reflect the gist of what I was after. Due to my own limitations, I can't say the I see the full picture still. Specifically, for the epjb article, for a particular combination of F and low driving frequency, an ensemble will exhibit ballistic flow. However, for any actual system, is it true that we would not know the phase Theta at which we are driving the system? So while there will be ballistic flow to either the left or right, we will be unable to predict which? If so, I could see how that knowledge might still be useful. Does my summary sound correct? –  lilinjn Aug 22 '12 at 12:22
    
regarding the step motor link -- Does this summary sound correct?: they propose a dynamical model for the motor, analyze the "chaotic structure" of the model, collect empirical data to confirm that the observed dynamics match the predicted "chaotic structure" and that it is of a relatively simple form, and conclude by asserting that "chaotic control methods" could be used which would require less energy than other more aggressive controls, without explicitly describing how those controls would work, but instead referencing other papers? (sorry for having to ask such basic questions...) –  lilinjn Aug 22 '12 at 12:29
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