The famous mathematician Pythagoras maintained that everything was moving in a cyclic way, like the celestial stars, so that finally, nothing changed. About the same time another sage man, Herakleitos of Ephesus, noted that everything flows ("Τ? π?ντα ρε?") and that you cannot cross the same river twice ("Ο?κ?νδ?ςτ?ν α?τ?ν ποταμ?ν ?μβα?ης"). Over two millennia later G. W. F. Hegel[1] tried to reconcile the two opposite ideas suggesting that everything was changing in a spiral way. J. G. Fichte[2] coined the terms thesis, antithesis and synthesis applied to the idea of Hegel. Thesis was changing to its opposite antithesis and this one to synthesis, the thesis of the next spiral. K. Marx[3], although starting from Hegel's ideas, suggested that "philosophers have only interpreted the world in various ways, the point is to change it". Thus he adopted the spiral evolution in history formalizing the historical materialism. Almost one century ago B. Van der Pol[4] described the relaxation oscillation that models evolution in a more accurate way in a physical, biological, psychological, social world. In the following the concept of relaxation oscillation will be briefly described together with possible applications on several levels.

Relaxation oscillation

Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value or between two or more different states. The simplest form of oscillation is the harmonic or sinusoidal one which continues indefinitely (if uninhibited by friction or any other dissipation of energy). It results when a capacity (or its reciprocal elasticity) is combined with an inertia. When the capacity is combined with a resistance, the relaxation oscillation occurs (Figure 1) Like The harmonic oscillation, it has an idiofrequency, the frequency that it is functioning when there is no force affecting it, but it is asymmetrical and consumes energy. The greater the physical size of the oscillator the longer its idioperiod is. It consists of two phases, one charging, slow, characterized by a negative feedback, and one discharging, fast, with a positive feedback. The change from the one phase to the other occurs at two predetermined thresholds respectively. During the charging phase a quantity is accumulated. When the threshold is achieved, a quality change occurs. Following the change from the positive to the negative feedback phase, a refractory period ensues which is initially absolute and proceeds into a relative refractory period. During the absolute refractoriness no stimulus may initiate a new oscillation, while during the relative refractoriness a strong stimulus may cause a new oscillation. Stimulus is the application (or removal) on a system of a small amount of energy so strong that it can cause a response. This response follows the all-or-non law, and continues although the stimulus has ended. If no stimulus appears, the oscillator will continue its charging course until the (upper) threshold is reached and the new oscillation starts spontaneously. It ends when the discharge reaches a lower threshold. Because of this responsiveness of a relaxation oscillator, its output may become the stimulus for another oscillator and this to another etc. Thus a wave is formed that may result in synchronization of the system that oscillates as though it were a single unit. The frequency of the system is close to that of the unit with the highest idiofrequency. When the idiofrequencies of the individual units are very close to one another the synchronization is easier (coordination). Depending on the idiofrequencies, the synchronization may become at a rate of 1:2 or 1:3 etc (harmonics). The refractory periods of the individual units do not end simultaneously in a system. A stimulus at a phase when some units have become responsible, while others remain still refractory will stimulate only the responsible ones. Subsequently the output of the newly stimulated units will trigger the rest while they have become responsible. Thus the system desynchronizes and the anomaly perpetuates itself. This short phase is a vulnerable one. Several phenomena that seem to be random are actually examples of relaxation oscillations since the signals that can stimulate them during the relative refractory phase may be random. Furthermore the thresholds may not be constant but vary affected by other oscillations. If the idiofrequency of the oscillation is high, a longer proportion of its period is refractory and the system is not easy to be excited by an external stimulus.

Figure 1: Relaxation oscillation.

Physical Level

I shall examine in a very vague way two natural phenomena, earthquakes and forest fires that are periodical. The earth rotation causes an elastic strain to the tectonic plates that constitute the earth's lithosphere. In spite of the strain, the existing friction resistance does not permit their movement. This phase may last for several decades.  The strain increases gradually, a charging quantity accumulates until a threshold is achieved. Then one plate moves releasing the stored energy and triggers the movement of other plates. Catastrophe ensues [5]. The next oscillation starts.

A forest consists of green leaves that cannot be burnt easily and wood that is flammable. While the trees in the forest are growing the proportion of wood over the green leaves is increased. An old forest has a lot of combustible vegetation. At a time when the combination exists of high temperature, drought and strong winds, as in summer time, the old forest is ready to be burnt. Any trigger may cause a fire. When an ignition source is brought into contact with a combustible material such as vegetation that is subjected to enough heat and has an adequate supply of oxygen (strong winds) from the ambient air a wild fire may occur. High moisture content usually prevents ignition and slows propagation, because higher temperatures are needed to evaporate any water in the material and heat the material to its fire point (threshold) [6]. Triggers may be e.g. lightning, self-ignition in landfills for waste, a spark between high voltage electrical wires, intentional or unintentional arsons etc. After a forest has been burnt new trees grow out of seeds that have not died been deep in earth. The next oscillation starts.

Biological Level

Many biological phenomena, both physiological and pathological, have a strong periodicity following the laws of relaxation oscillators. The woman's period is an example of normal periodicity. Intermittent pyrexia as in malaria is an example of pathological periodicity. Almost a century ago Van der Pol[4]himself constructed the first model of the heartbeat. It consisted of three electrical relaxation oscillators and a triode lamp. Each oscillator had at least one capacity, one resistance and a neon led (light-emitting diode). Almost half a century later a new model was constructed consisting of seven Van der Pol oscillators without using a triode [7], [8]. This model reproduced the cardiac rhythm with all known rhythmic phenomena of the heart, both normal and abnormal, except fibrillation that needed many more oscillator units (Figure 2, Figure 3). this model suggested that the re-entry phenomenon, a known mechanism of cardiac arrhythmias, did not need a cyclic movement. Like the heart, it also, had a vulnerable phase and a strong stimulus upon it may desynchronize the system resulting in sudden death in humans. It also manifested rare dysrhythmic phenomena that had not been observed in the heart. They were looked for in clinical cases and it was shown that they actually could occur in the human heart.

Figure 2: Analogue of ectopic beats in the form of bigeminy. When the basic frequency increases the ectopic beats vanish.

Figure 3: Analogue of absolute arrhythmia as in atrial fibrillation.

Psychological Level

The human "psyche" i.e. the aspect of Ego conceived only by itself [9], has an input (logic), an output (volition) and a connecting element (emotion). It also exhibits some kind of automatic periodicity. Logic is formed out of signals introduced from the physical and social environment through our senses. Volition leads to signals, movements or secretions that may affect the environment. The output may be initiated by logic as a deliberate desire ("προα?ρεσις", Aristotle [10] or spontaneously in a periodic way as an appetitive desire ("?ρεξις", Aristotle [11]  A painful stimulus causes a movement that tends to remove the stimulus and this is an example of optional (deliberate) response.  An example of periodic volition (Figure 4) is the need for food. Hunger is gradually increased until it reaches a threshold and causes movements and secretions that lead to finding food and eating. This bodily response is a positive feedback until saturation occurs and eating stops.  Saturation means absolute refractory phase. During this phase the subject wants to not eat. Gradually the absolute refractoriness evolves to relative refractoriness and the subject does not want to eat, so that it does not start the procedure of taking food. However, if a proper appetizer appears, the threshold is reached prematurely. Thus the phases are: I want not to; I do not want to; I want to; I do eat up to saturation and again I want not to.  A similar series of events appear with libido. The phases are now: I want to make sex; I do make sex up to orgasm; I want to not make sex for some time; I do not want to make sex, but, if I am triggered e.g. by seeing a naked provocative body, I proceed prematurely to make sex; if not, as time passes by, I come to a threshold and I want to make sex and I do, either with a partner or by self-satisfaction or even by a nocturnal emission. Note that "I want to not..." and "I do not want to..." have not exactly the same meaning. The former means that I object to do something, while the latter means that I do not do it spontaneously but, if I am triggered, I do it.

Figure 4: Volition oscillation and its phases.

Social Level

I shall discuss the oscillating nature of economy and politics. Normally there is a negative feedback between production and consumption. Whenever production (and supply) are less than consumption (and demand), the prices go up and both the consumption is reduced and the production is increased, since the high prices are a motive for producers and suppliers [12]. The opposite holds whenever production is higher than consumption. As the price of the goods rises, the quantity offered usually increases, and the willingness of consumers to buy the good normally declines. The prices fall and equilibrium is established at a price that varies very little. This equilibrium is the basis of liberal economy that is self-regulated. A similar mechanism applies in wages. When the demand for workers is increasing relative to the supply, the salaries also increase and vice versa. This way the economy is booming. Gradually production increases overwhelmingly. Even though the products become very cheap, they cannot be sold. The producers are obliged to lay off workers. These people cannot buy anything and the consumption is further reduced. A vicious cycle, a positive feedback, occurs and this is an economical crisis. Such crises occur periodically and may be transmitted as a wave from one country to the other and become international. An economical crisis may also occur whenever the supply is reduced to a very low level as e.g. during a war or an embargo. The consumers hasten to buy goods as long as they still exist in the market and the lack of them increases, together with the demand. Another positive feedback results, economical crisis occurs. In both kinds of crisis the intervention of the state may stop the vicious cycle depending on the situation.

Society changes continuously. Its volition also changes and this is the basis of ethics. Ethics, however varies and is subjective so that an objective law is needed. While society changes a gap develops between the ever changing social volition and the constant law. People protest, the state applies suppression. The people protest now not only because of the law-ethics gap but also because of the suppression against them. They too apply violence. Violence begets violence. This is a positive feedback, a vicious cycle. When their difference accumulates to a great extent a major political change (crisis) occurs, as e.g. revolution.

Applications

Each unwanted situation needs proper treatment. However the laws of relaxation oscillations are common to all periodical phenomena. Thus, an oscillator system is synchronized at a frequency close to the idiofreqency of the fastest oscillator. The higher the frequency, i.e. the shorter its period, the higher becomes the ratio of the refractory phase to the period. Thus the system renders itself less vulnerable to accidental external stimuli. This "cure", however, has limitations. Unlike the sinusoidal oscillations, the relaxation ones consume energy. If the cause of abnormality is lack of energy, the situation may deteriorate when working at a higher frequency. In the heart an increase in the cardiac rate reduces extrasystoles, some kind of arrhythmias, but if the cause of the arrhythmia is myocardial ischemia, i.e. deficient blood, oxygen, energy, supply, the outcome may be ominous. A political change by preprogrammed elections is followed by a refractory period, a grace period for the new government during which it can apply necessary unpopular measures without violent resistance. In republics, the bigger a state (the "oscillator") the longer the period may be between two successive elections. If there are frequent political crises, more frequent elections may contribute to a more stable political situation. The periodical change of government by elections in a republican system, is some sort of controlled crisis that prevents uncontrolled ones by reducing the period of the oscillation.

Up to now the laws of relaxation oscillators have not been applied on physical phenomena. However, it might be useful to examine if they can be applied. For example, controlled forest fires during winter time, when a wild fire is impossible, may clean up the forest of combustible material, and reduce the probability of uncontrolled wildfires in summertime. Similarly, in quake prone areas it might prove useful to produce, if possible, a controlled quake e.g. every 15 years (i.e. before the anticipated next crisis), so that the accumulated energy be dissipated. The controlled quake with reduced intensity, at a given place and time, is not dangerous.

Periodicity occurs in all levels of organization and usually takes the form of relaxation oscillation. Analyzing such phenomena in terms of oscillations may help in gaining a better understanding of their mechanism and finding ways to correct abnormalities in their function.

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