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Updated 20 September1998

A. Dughin

Lyapunov Time

In modern physics, primarily exploring “very unbalanced conditions” and chaotic systems, there is a technical term - “Lyapunov time.” It designates a period, when a certain process (physical, mechanical, quantum, or even biological) moves beyond the bounds of precise (or probabalistic) predictability and enters a chaotic mode. In other words, the tajectory of the process is subordinate to strict laws only up to a certain moment in real time. Beyond this moment, “normal” time ends and paradoxical “Lyapunov time” (or, more precisely, “positive Lyapunov time”) takes over. The characteristics of this “time” are very curious. Unlike usual physico-mechanical time, which is regarded in classical physics as an essentialy reversible quantity (this means that time is nothing else but a static axis, adding a fouth dimension to three-dimensional space; refer to the educational Einstein model), “Lyapunov time” flows irreversibly, only in one direction, and, consequently, consists not of a once-and-for-all defined trajectory (in four-dimensional space), but of “events,” completely unpredicatable movements, which are arbitrary, accidental, irregular. Processes that occur during “Lyapunov time,” are called chaotic in contrast to processes in classical mechanics.

This can be illustrated with an example from everyday life. For instance, three people are drinking. Until a certain moment, their behavior is quite predictable: they discuss aquaintances, friends, personal problems, sports, women, politics. Gradually, as the level of intoxication increases, “noises” (this is how modern physics calls insubstantial interferences in the flow of the process)begin to creep into the conversation. These “noises” may be expressed in that certain passages are repeated by drunken people several times, the psychological conditions become tense, arguments and conflicts erupt, the general atmosphere tightens up. At a certain moment, conditions reach the point of bifurcation (this is a key term in the “catastrophe theory” of the well-known physicist Rene Tom). This means that the logic of behavior of the drunken trio as a whole, and of each of its members separately, can arbitrarily take one of two trajectories of equal probability. For example, two of them go to sleep, and the third goes home. Or, one assaults another with fists, while the third calms them down. Or, all three move into the street and scuffle with passers-by over trifle matters. Or, all peacefuly disperse and come back to their families with a guilty conscience.

When everybody gets together to drink, the final result of the booze-up is not known. Up to a certain point, the situation is subordinate to a limited number of psychological factors, varying depending on the cultural and intellectual level of the drunks. But, whatever the preconditions, if the drinking spree progresses, sooner or later the point of bifurcation is reached, and the group unnoticeably enters into “Lyapunov time,” where all proportions are eroded, where a minute detail may cause an inadequately massive reaction, where every following action is completely unpredictable and lacks motive.

But what is most interesting is that “Lyapunov time” is not a period of complete disorder, where all movements are absolutely arbitrary. It is something in the middle between a fully structured system and a complete absence of a system. Scraps of trajectories remain, drunk behavior is subordinated to fragments of logical-psychological determined strings. Chaos has its paradoxical structure, which is called “physics of unintegrated processes” or “system of fractal attractors.” Consequently, “Lyapunov time” is subject to a certain paradoxical measurement, only more flexible and widely understood than determinism of “integrated systems” (in other words, ordinary classical and quantum trajectories). Some contemporary physicists - in particular Ilya Prigogine, believe that processes flowing in “positive Lyapunov time” are the key to the mystery of life. Here, in this transitional state, between strict structure and complete absence of structure, in a chaotic system lies the “magic” combination of law and freedom, model and event, determination and spontaneity, and it is this combination that is called “life.”

A purely logical, rational model, as shown by Kant, is not capable of “grasping”an object in itself, gist of reality, which always remains inaccessible and noumenal. The “noumenon” itself keeps complele silence. Only in chaotic worlds, during “Lyapunov time,” occurs the secret transition from silence to speech, from existence to nonexistence, from rational to irrational, and in reverse.

Amazing, but the ideas of Prigogine and other theoreticians of “unitegrated processes” strictly concur with traditional doctrines of alchemy, holding that the “Philosopher’s Stone” can be found in a “particle of ancient chaos,” which the creator neglected during creation! This is the “philosophers’ magnesia,” “our Cibela,” “our Latona.”

“Lyapunov time” is a very important concept on two isomorphic levels - individual spiritual realization and social transformations. For an individual, trying to find its true center, “Lyapunov time” means cultivation of borderline states, between fresh daytime consciousness and nightly (alcoholic, narcotic, etc.) fainting. Only on this brink is it possible to grasp the magical, spectral point, where individual existence borders with extraindividual realities - of infracorporal, as well as purely angelic character. This is the essence of the machanism of initiation. “Lyapunov time” is a phase of “initiational death.” One who is able to take control of this “bottleneck,” goes over the brink of fatal dualism life-death. On the social level - an analogous picture. Every regime, social arrangement, economic-political formation is subordinate to strctly determinational laws, embodied in the power structure, its ideology, its internal standards. But social energy, like any energy in the corporal universe, diminishes in one direction, “produces entropy.” That is why any regime and any social formation function logically and according to laws only for a limited period. After a certain moment, “Lyapunov time” arrives. Like a group of drunks, upon reaching a certain boundary, society starts to behave unpredictably, chaotically. The peripheral grows to giant proportions, central, axial moves to the margin.

Unquestionably, “Lyapunov time” began for the USSR in 1985. The current president (note, “unpredictable”!) is a typical example of a “particle” of a chaotic system. In front of our eyes, a new liberal system is born from the “dissipating remains” of late degenerate socialism. But it, too, grows older, its entropy increasing terribly fast, it begins to, amazingly, to the details, remind us of last phases of the Soviet society. It cannot be ruled out that the liberal cycle will be very transient, since some systems are, in principle, not viable (in certain conditions).

Another important aspect: the phase of collapse of Sovietism occured with complete intellectual passivity of major players. In other words, there is no social organism, which could “grasp” the main content of social “Lyapunov time” in our situation and put this precious knowledge in the basis of a new social order. It appears that everybody slept through the most interesting events. But initiational death is different from ordinary death in that consciousness is not lost completely (being preserved in a special regime). Chaos must be not only outlived, but interpreted. If this does not occur, repetition of chaos is inevitable. Another catastrophe, another phase of social changes, yet another chord of the “dissipative leap.” More than that, this will repete (with a faster rhythme) until some social formation takes on the responsibility for a dangerous and exciting scientific and practical work with chaotic structures.

Contemporary “stability” and “steadiness” are even more ghostly and deceiving, than the last days of Sovdep (and return to the past is unrealistic).

Our society today - is as much an incorporeal mirage, as the self-confident idiocy of the contemporary man in the street. But we know that “Lyapunov time” is our time. That is why, the hand is stretching for...(no, not to what you were thinking, yet) books by Poincare, Kolmogorov, Stengers, Tom, Prigogine, Capra, Nichols, Mandelbrot, and other interesting authors.

To our universal doctrine of Revolution, apart from heritage of “New Right” and “New Left,” we add the theories of “new physicists.”

Translated by Victor Olevich

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