Basic ideas and concepts
A delayed action, is an action which actually occurs at a later date than initially expected. The corresponding time shift will be called &tauD.
But the time t is mathematically like a spatial coordinate, and the quadrivector of relativity considers it as is, since it is used as the 4th coordinate after the "real" spatial ones, x, y and z.
So, the time delay &tauD is not necessarily a bad event. You are usually blamed when coming with a time delay at an appointment, but one excuse you could give, is that you actually took benefit from this delay, to fill this "spatial" interval with interesting informations (but you could also fill it with nothing, meaning then that you lost your time while losing the opportunity to fill this temporal space you have actually created for nothing).
A time delay can thus be considered as something positive, as soon as you are able fill to this time interval with important, or relevant, or usefull events, and of course if you are able to use these events in the future. Such a time delay is thus associated to the memory of these events, which can play a crucial role in your history, compared to the situation for which no delay would have occured.
A delay can thus consist of a positive notion, as soon as it is filled with some events which can enrich your history compared to the same history, but without delay.
A delay can also be simply regarded as the time flow itself, since being late, you did not create any time interval, you just shifted the chronology of your history (and the one of the people who were waiting for you).
A "constructive" delay is a kind of memory space (time interval) which is filled with interesting knowledge you can later use (because you remember it). The more important is the delay, the more complex and rich is your knowledge. Delay or memory, is like a library filled with interesting books. The more you have books (and the knowledge,a nd understanding of their content), the more complex can be your actions as a smart use of the accumulated knowledge. The complexity of a delay dynamical system is proportional to the size of the memory, or to the size of the delay, which is filled with interesting information. Here comes also the notion of action, or also we could say reaction. Indeed, it is useless to accumulate information and knowledge, if one never makes any practical use of them (reading books only your whole life, is not something recommended to properly fill it with satisfactory content; you need to write your own books, or use the knowledge you learnt to fabricate something, and fill thus your own life also with positive (re-)actions, which can be usefull to fill the others' delays).
There are then different possible (re-)actions. The linear one is a kind of copy, understand, and past. It is an action, so it is already good in some sense. But there is (according to me), a potentially much better one, which we will call the nonlinear way. The nonlinear way is potentially capable of creating new informations (which will be able to lead to new books), novel knowledge.
Here comes then the hand-waving viewpoint I could propose for a (nonlinear) delay dynamical system: you accumulate a succession of temporal events in a memory of a finite size, and you use it (if possible in a nonlinear fashion), in order to create a new succession of events over the same time duration corresponding to the delay memory. The practical image of this viewpoint could be the story of an everlasting day (day = &tauD), in which you wake up every morning on the same day, but you do have the memory of what happended the same previous day. Even in that limited spatio-temporal world (which could be referred as a closed loop system), there are an infinite possibilities of living the same day. This configuration is however, much more limited than real life. Limitations come from the fact that it is the same day, which is fortunately not the case of real life.
A more physical sense of the dynamical limitation in delay systems appears at that point, which is the typical time scale at which you can react within this day. The reaction time is limited by the maximum speed of your movements, of your speech rate, of your interaction with other people, and so on ... The time scale of this limitation is typically of the order of a second. The actually limited complexity of a delay dynamical is roughly given by the number of reaction time scale, you can fill in the total delay (thus of the order of 24 x 3600 ~ 100 000, which is the order of the temporal dimensionality of the possibilities one would have in such a closed loop everlasting single day life).
As a conclusion, I should clearly state that this modest introduction to delay dynamical systems is from far not a tribute to the attitude of being late. Moreover I definitely think it is not a polite way to behave with others, and as a humbly apologise, I would dedicate this introduction to all the ones who had to experience a delayed appointment due to one of my late arrival, ensuring them that I definitely tried to do my best in order at least to fill properly this delay time interval with some constructive motion.
Practicals example of delay dynamics
- The vertical station of a human being (delay = transmission time from the sensors -eyes, or equilibriumin the internal ear- to the brain and then from the brain to the reaction actuator -the feet-).
- A stick pointing upwards, and which vertical position is controlled by the end of a finger (so called inverted pendulum, see videos on John Milton's website, same delay and reaction times as in the previous example).
- Population evolution (delay = one generation; reaction = death and birth rate).
- Distant control of a satellite from the earth, or any far away object (delay = back and forth traveling time of the signals from the object to the control center, which can be as high as several milliseconds; reaction = response of the electronic / motion actuators of the object). As far as I heard, the space conquest was one of the strongest motivations of a strong theoretical and scientific interest in the history of delay dynamical systems.
- Many physiological systems, among which the most popular in the academia might be the blood cell production model (Mackey-Glass model; delay = transport of the information from two different places in a living system; reaction = concentration sensing and production rate). But also many other physiological example are concerned a priori with a delay dynamical modeling, gene transcription, hormones production, etc...
- Cutting turning machine (delay = one round of the machined object; reaction = mechanical interaction between the machined turning object, and the cutting tool)
- Laser diodes subject to optical feedback (delay = roundtrip time between the laser output facet and the external reflecting surface; reaction = photon and carrier life time of the laser diode)