4. Critical Constellations

Jiu-jitsu against wrestling or do the right little thing in exactly the right moment!!
Normal engineering (and also medical) technique is to make a system behave in a given way by applying brute force. In the presence of multistability (e.g. if chaotic and stationary modes are both stable with unchanged parameters), a chaotic waterwheel can obviously be forced to change to the stationary mode by overriding its internal dynamics with a larger external force that implies the correct constant angular velocity for a certain time. The more elegant way is to give it a small kick in exactly the right moment and let its own dynamics perform the work. Such sensitive moments are well known: In economy they are called "windows of opportunity"; psychologists use the term "kairos" for the critical moment during a psychiatric treatment, where the patient sensitively reacts to the treatment ("kairos" is the Greek word for "time"). That such a simple system as our waterwheel displays that rather psychological behavior is a surprise!

Hints for demonstrations

By selecting Critical Constellations after stopping the waterwheel and starting it again, it will move chaotically. Please observe during about a minute that chaos does not stop (it will not stop even after hours). Try to intuitively understand when you have to apply a "kick" to cause a phase transition into the stationary state. I guess you will not have any success! Try it again by applying the "kick" at EXACTLY time 20 s. WOW!!! As the sign and amplitude of the "kick" are adjusted to this critical constellation, you will probably not manage to cause a transition at the next critical constellations occurring at 52 and 57.5 s (see below).

Another way to accelerate chaos-order transitions is to apply torque noise of optimal amplitude (not implemented into this demonstration). This before not known noise effect was published in Physical Review E, Vol. 55(3), 1997, p. 2215-2221 by F. Gassmann under the title "Noise-induced chaos-order transitions". The respective procedure has the advantage that the critical constellations have not to be known, though noise is only effective within the critical windows. The disadvantage is that transitions occur with probabilities below about 13% only. The noise-induced transition probability e.g. within the above mentioned window at 20 s is 1.8% only. Each window is around 1 s wide and they follow each other in irregular intervals. In the average, these intervals last about 100 s. With the specially prepared initial conditions, the first critical constellation occurs at time 20 s, the next at time 52 s followed by one at 57.5 s. Until time 400 s, a total of 4 critical constellations occur (between 200 and 400 s there is a wide gap without critical constellations).

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