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Do you know cases where chaos theory is actually applied to successfully predict essential results? Maybe some live identification of chaotic regimes, which causes new treatment of situations.

I'd like to consider this from the engineers point of view. By this I mean results that are really essential and not just "interesting". For example one might say weather calculation effects are explained with chaos theory, but an "engineer" might say: "So what? I could have told you without a fancy theory and it provides no added value since there is nothing I can change now."

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I think it depends on the meaning of "application". In the wikipedia entry there are a lot of applications of chaos theory listed:

Chaos theory is applied in many scientific disciplines, including: geology, mathematics, microbiology, biology, computer science, economics, engineering, finance, meteorology, philosophy, physics, politics, population dynamics, psychology, and robotics.

For example, in engineering there is the following in the abstract of a paper:

Control of chaos: Methods and applications in engineering ☆A survey of the emerging field termed “control of chaos” is given.

Several major branches of research are discussed in detail: feedforward or “nonfeedback control” (based on periodic excitation of the system); “OGY method” (based on linearization of the Poincaré map), “Pyragas method” (based on a time-delay feedback), traditional control engineering methods including linear, nonlinear and adaptive control, neural networks and fuzzy control. Some unsolved problems concerning the justification of chaos control methods are presented. Other directions of active research such as chaotic mixing, chaotization, etc. are outlined. Applications in various fields of engineering are discussed.

Chaos theory is not chaotic in the everyday sense, there are those strange attractors and the behavior of collective solutions to dynamical equations, which is what chaos theory deals with, may very well give a handle to control complex systems. It seems to still be at the exploration and research phase, but yes, there already seem to be possible applications.

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I don't have a very detailed list of applications, but off the top of my head, I would say the Stability of Solar System is an essential prediction. Yes, I understand that it is not extensively essential, because I can't imagine what we would have done if, on paper, the System turned out to be unstable. As far as the applications of Chaos Theory go, it is more about controlling the Chaos that make more sense. How much noise should you add to the system, how strongly should you couple two systems to make them predictable? These are some generically important question that Chaos Theory tries to answer.

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