Chaos theory

Chaos theory. Chaos theory is the mathematical theory that deals with systems with unpredictable and apparently fortuitous behavior (although their behaviors are governed by strictly deterministic laws). Formulated as the alternative to the classic one of the turbulence, it is based on the mathematical study of the dynamic systems, since its beginnings in the 70’s of the 20th century , it has become one of the most developed fields of research. Today it is considered the third revolution in physics after the theory of relativity and quantum mechanics .

Summary

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  • 1 History
  • 2 Best known contributions
  • 3 Balance of Chaos
  • 4 Dissipative structure examples
  • 5 Charts
  • 6 Other contributions
  • 7 Presence of Fractality
  • 8 Approaches to chaotic systems
  • 9 Technological applications
  • 10 Relationship between order and Chaos
  • 11 Source
  • 12 References

History

In the ancient theory of creation, Chaos was the dark and silent abyss from which the existence of all things came. Chaos gave birth to the black night and Erebo, the dark and unfathomable region where death dwells. These two children of the primitive darkness were united in their turn to produce love, which originated light and day. In this universe of formless natural forces, Chaos generated the solid mass of the earth , and from which emerged the starry, cloud-filled sky. In later mythology, Chaos became the formless matter from which the cosmos or harmonious order was created.

Best known contributions

One of the best known contributions to Chaos theory has been that of Llya Prigogine . Its dissipative structures are based on a deep review of thermodynamics , which stops dealing with systems that are close or in equilibrium, to study those that are far away. The existence of Chaos can be observed in a great variety of systems, in the world of fluctuations, chance and bifurcations , multiple times and dissipative structures. In short, the world of living beings, if it is defined as an open system far from equilibrium.

Until now science has dealt with systems on predictable principles, at least on a large scale; however, the natural world shows tendencies to chaotic behavior. For example, large meteorological systems tend to develop random phenomena by interacting with more complex local systems. Other examples, the turbulence in a rising smoke column, or the human heartbeat.

For a long time scientists lacked the mathematical means to deal with chaotic systems, they tended to avoid them in their theoretical work. Starting in the 1970s, some physicists began looking for ways to deal with Chaos. One of the leading theorists was the American Mitchell Feigenbaum , who determined certain recurring patterns of behavior in systems that tend toward Chaos, and involve constants now known as Feigenbaum numbers . They are related to those shown in fractal geometry and the study of chaotic systems that have affinities with catastrophe theory.

Balance of Chaos

At or near balance, nothing new can emerge. the difference is drowned late or early, with no new order emerging.

Far from balance, the situation is much more complex and rich. Imbalance phenomena begin to occur as a source of order or order through fluctuations called dissipative structures.

The equilibrium structures can be maintained by means of reversible transformations that imply small separations with respect to it. Dissipative devices are completely different, they are formed and maintained through the exchange of energy and matter in the course of an unbalanced process.

Dissipative structure examples

Among the simplest examples are the so-called Bénard instability. Derived from hydromechanics, it shows the appearance of order in an open system far from equilibrium. Imagine a container with liquid inside. If you evenly heat the bottom, we will first be in the linear zone and the heat is transmitted by simple conduction through the liquid. As the temperature rises and a high gradient is reached, the heat will be transmitted by convection, on the surface of the molecules they will be arranged to produce cells similar to those of a honeycomb, while the energy continues to be supplied, the order of the molecules will continue to be simple. view.

This happens when the system reaches a certain threshold, the fluctuations that were previously canceled, are now amplified, giving rise to a macroscopic current whose effect is the appearance of a new order in which thousands of small units cooperate.

This is a very simple case and only allows you to associate the system in an order type with its persistence time. In other complex associations of chemical reactions, bifurcations corresponding to different types of states are presented, he must choose a branch of the bifurcation and one of the times of multiplicity.

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When graphically representing repeated functions, it is very common to observe unpredictable results with incredible sensitivity to the initial parameters. For the study of the chaotic behavior of these nonlinear functions, bifurcation diagrams are applied that represent the change of the result of according to the initial parameter.

From these experiences Prigogine concludes that chance and determinism cooperate in the construction of change. In a bifurcation, it depends on the chance that the system evolves in one direction or the other, it is the randomness of the fluctuation. The curious thing is that between one and another bifurcation, the hegemonic lies in the determinism expressed in the differential equations. Here chance and recurrence are not opposed, on the contrary, they come together to trace the history of the system.

Other contributions

Edward Lorenz used a computer program to calculate using various equations of probable weather conditions, but he understood that by rounding the initial data, the ends were different. He discovered that it is due to the feedback loops and reiterations of the chaotic system that the atmosphere represents. Lorenz had intuited the butterfly effect. Accordingly, it is not difficult to think that perhaps after multiple feedbacks and forks of the system, a butterfly with the flap of its wings could produce a tornado on the other side of the earth.

In 1963 Lorenz manages to compute the first strange attractor and the first to discover its fractal dimension, called Lorenz. It was characterized by being stable, of few dimensions and not periodic. Their sockets did not intersect each other, their loops and coils were infinitely deep; they never met or intercepted each other. However they remained within a finite space, confined in a box.

The theory was summarized by Lorenz simply and deeply as, “Chaos: When the present determines the future, but the approximate present does not approximately determine the future.” [one]

Presence of Fractality

Fractality is characterized by chaotic systems or phenomena, which may appear to be random phenomena, but they are not, they present a non-linear dynamics (sensible dependence) and their behavior, taken to the phase space, is characterized by the existence of attractors strange, fractal dimension.

Approaches to chaotic systems

In chaotic systems two general basic approaches are presented: Chaos as a precursor and partner of order and not as its opposite. A key figure in this research direction is Llya Prigogine, a first prize for her work in irreversible thermodynamics. Secondly represented by EN Lorenz, Mitchell Feigenbaum, Benoit Mandelbrot and Robert Shaw … who highlight the hidden order that exists within chaotic systems.

The approach of the strange attractors differs from the order from the Chaos by the attention that presents the chaotic systems, that is to say those that do not generate order. While Prigogine considers that the order line from Chaos (philosophical line) resides in its ability to solve the metaphysical problem of reconciliation between being and becoming, the defenders of strange attractors highlight the ability of chaotic systems to generate information . Order would be nothing more than a manifestation of disorder, an anecdote.

Technological applications

Chaos begins to have technological applications, due to the engineers’ consideration that if it exists in nature, it should be used and not be afraid of it. Some groups have already found a way to produce synchronized Chaos signals and use it in communications. Applying this type of ideas economic problems can be studied. Non-linear computer programs have been made that simulate social phenomena and have been able to reproduce bubbles and collapses. There is a perfect equivalence between the dissipation required by the fluctuation order model and social instability. Chaos is applied in the behavior of the stock market the theory of price formation, modernization of an industrial plant, trade between two countries and global knowledge.

Relationship between order and Chaos

In reality there is no order without Chaos that generated it, nor order generated without Chaos that regenerates it. The only natural curves are fractals.

Nature has no origin, it is always being born. For there to be zero time there would have to be a universal clock, and Einstein forbids it. Each person or thing is a clock. There are many times: the linear ones (that of the thermodynamic date or the irreversible fall of any system of transformation of energy into heat); the historical (the irreversible ascent by the scale of complexity).

There are circular times, from whirlwinds or planetary systems to biological cycles. Tangential: the minimal random disturbance that starts the genesis like the Peking butterflies, where the regenerations, mutations, errors and background noises, their tracks really constitute the information.

There is no information where there are no differences. The combination of these elemental components accounts for figures of time. What we understand in order are just vacuoles of meaning floating in Chaos. They come from Chaos and return to it.

 

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