A good, concrete example of using "chaos theory" to solve an easily understood engineering problem?

A stepping motor 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

The chaos theory enables 2 broad families of engineering applications.

• First bases on the fact that low frequency periodic unstable orbits are embedded in each chaotic attractor. In other words there are very simple and periodic dynamical modes inside the complex chaotic behavior. 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 basins of multi-stable systems. 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