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The control of chaos

The project even created a forecast model for the stock market

There is something in common between the fluctuations of the share values negotiated on the stock market and the distribution of the nucleotides in a molecule of DNA, between the orbits of the planets of the solar system and cardiac rhythm, between the workings of industrial tools and the transmission of communication signals. What brings these so diverse phenomena together is their natural state of chaos. Today it is known that chaotic phenomena are not just predictable to a certain point, contrary to random chances which are completely unpredictable, but also potentially controllable. In this manner, chaos could be an ally that allows the human being to exercise more freely his creativity. The potential applications of the control of chaos has spread through areas such as the financial markets, telecommunications and mechanical and genetic engineering.

In the present front line of the applications of the theory of chaos is the Brazilian physicist Dr. Celso Grebogi, who had previously spent two decades in American research institutions and had published close to 250 articles on the theme, in magazines such as Nature, Science and Physical Review Letters. In March of 1990, Grebogi and his colleagues Edward Ott and James Yorke, from Maryland University, revealed the possibility of controlling chaos. The impact of the idea, published by the Physical Review Letters, was immense. Within only a few months, experimental groups from other institutions had passed from the idea to the practical in various fields.

Now, at the Physics Institute of the São Paulo University (Ifusp), the native of the state of Paraná Grebogi is part of a team with the veterans Iberê Luiz Caldas and José Carlos Sartorelli, along with youthful Murilo da Silva Baptista, that is developing a thematic project on chaos. The contributions of the group include a dynamic model of the financial market, the fundamental theories of a system of communication, an experimental study of the formation of bubbles destined to improving industrial processes and a dynamic model of mixtures, whose possibilities run from the creation of plankton to the production of paints.

The development of chaotic dynamics allowed for something considered to be impossible, to predict the behavior of the stock market index. The trustworthiness of the program done by the USP team extends for a period of up to two months (see graph on the following page).

“The oscillations of the stock market arealmost at the edge of random chance. They are, as people are accustomed to saying, totally chaotic or super chaotic”, states Baptista. “It is a turbulent system, with lots of variations. It is impossible to understand all of them, but all together its recurrence is typical of a system with few variables”. For this reason, the physicists didn’t intend to determine what would be the value of each individual share, but managed to forecast how long the system as a whole needed to come back to its original state.

Caldas completes, “The average time for the shares to present a determined gain or loss is related to the amplitude of the oscillations through the same type of statistics that allow for estimating where a determined nucleotide will reoccur in the DNA chain. We are dealing with a Poisson distribution, a function of probability that determines the frequency of an event, something well known by mathematicians.”

Chaotic communication
The area of telecommunications is another in which the use of chaos is highly promising and in which the Ifusp team is carrying out theoretical and experimental work that is highly advanced. “With a power oscillator, it is produced a chaotic wave”, describes Dr. Baptista. “Through small disturbances, it is possible to make this wave remain above or below a certain level. By convention, if it remains above this signifies a ‘1’ and if it remains below a ‘0’. In this way we have everything we need to establish binary communication.”What is the advantage in this? Baptista answers: “In traditional linear communication, the loss of a part of the wave is irrecoverable, since it is impossible to know if you lost a ‘0’or a ‘1’. In a chaotic system this does not happen. Each state of the system depends on the preceding state, in such as way that at least as a probability, it is always possible to reconstruct the lost material.”

With the chaotic wave, he states, all of the demands of protocols (stages) of communication are carried out simultaneously, while for the same result with linear waves the process is carried out sequentially. Besides this, the production of the chaotic wave demands a minimal amount of energy, as its use increases significantly the capacity of the channel, because it allows for transmission on a line full of noise as if it didn’t exist. This is a form of natural communication that characterizes, for example, an exchange of ions between cells.

Drops and bubbles
On another research front, Dr. Sartorelli is redoing, with refinements, an experiment with drops created during the 80s by Robert Shaw, of the Santa Cruz University in the United States. Shaw studied the drip from a water tap and, on noting the time interval with which the drops fell, verified that it was a chaotic phenomenon. Dr. Sartorelli has assembled a refined apparatus, a water tank, cylinders with a mixture of water and glycerol and a monitor, which has been showing him how the drops and bubbles form and has proven the existence and behavior of chaotic systems.

From that point on, he studied the formation of air bubbles in a liquid column and deduced a mathematical model of wide applications. It could help in the control and the quality of petroleum derivatives, with improved efficiency in the distillation of the naphtha. In the process of the fermentation of drinks and food, it would allow for an accentuation in the taste of the gaseous drinks or for obtaining a greater yield in bread preparation. By helping the prevention of the formation of bubbles in bloodcirculation, it would allow divers to go deeper and to remain down for a longer period of time.

Blessed chaos
It’s not always of interest to control chaos. Sometimes it is more advantageous to provoke it. An example is the control of epilepsy. The epileptic state is a periodic phenomenon through which, recruited by a center, more and more neurons begin to act in synchronization, which is abnormal. In such a situation, which can damage the nervous system, the brain uses mechanisms that can break down this synchrony and recover the normal chaotic working. This is also the function of the anticonvulsive medicines. For the USP team, it is possible to obtain an analogous effect if, instead of betting on the inhibition of the neurons, one communicates to them a small additional stimulus, capable of disorganizing their uniting activity and re-establishing the chaotic system.

The neurological use of induced chaos is still speculation, but in the United States the work of Grebogi’s team inspired a valuable experiment. Researchers at the University of California demonstrated that one can use the control of chaos to direct cardiac activity. By means of a drug, they induced chaotic heartbeats in the hearts of rabbits, and with small electrical impulses, brought the hearts back to their normal periodic state, thus manipulating this condition. On their command, the hearts beat more quickly or more slowly. One of the possibilities opened up by this experiment is that of implanting a device beside the heart to detect whatever anomaly and to get it back to periodic beating. “In this manner, we would have an intelligent timer”, imagines Dr. Grebogi.

The extraordinary potential of a chaotic system comes from its versatility. Beginning with a given configuration, one can evolve in many manners. An example is the DNA molecule, combining only four nucleotides which can form millions of distinct proteins.

Hidden order
Dr. Grebogi and his North American colleagues demonstrated that one can direct the course of chaotic systems through small interventions. Instead of fighting against chaos, one takes away a part of it. “We chose one of the possible developments of the system and we induced it with adequate stimuli”, he sums up. This only occurs because there is always a hidden order in chaos. It is important to underline this point, since common sense is accustomed to confusing chaotic phenomena with random phenomena.

There are three types of systems, periodic, random and chaotic. The periodic systems are those whose evolution is totally predictable. This is the case of a simple pendulum, knowing some parameters, one can tell its position at every instant.

Then the random systems evolve in a totally unpredictable manner. It is pointless knowing the initial state and impossible to know what will be the subsequent state as it occurs with a throw of the dice that could result, with equal probability, in six distinct numbers.

The chaotic systems place themselves between these extremes in the vast territory that separates periodicity from complete chance. Their evolution is only foreseeable in a restricted interval of time. From that moment on, their high sensitivity to small disturbances makes forecasts impracticable. This is the case of Hyperion, one of the moons of Saturn. “If we were to travel to it, we would never reach it.” said Dr. Grebogi. “as time passes, the discrepancy between the forecast position and the real position grows exponentially.”

Small disturbances
Since its replies are not proportional to the stimuli, chaotic systems arecalled non-linear, the non proportionality impeding that the relationship “response against stimulus” be represented graphically by a straight line.These systems divide themselves into two groups, one is that of natural chaos (brain waves, the trajectory of planets and the fluctuations of the financial market), and the other that of the systems which only become chaotic when they are stimulated. “The same system, such as the dripping of water, can have periodic or chaotic behavior”, says Sartorelli. The non-linear function between stimulus and response explains why, in car plants, the robots work slowly. Whatever attempt to increase the velocity introduces into its movement a non-linear term that makes it stray from the established pattern and make a mistake in the work.

It is exactly this sensitivity to small disturbances that allows for the control of chaotic systems. It is enough to use the disturbances intelligently to make the system respond in the manner that is desired, as it has already been done in the United States for some time. “The tool that cuts sheets of aluminum to make soft drink cans possesses a chaotic movement that tends to produce incorrect cuts”, says Grebogi. “However, with minimal stimuli, one can eliminate the undesirable vibration from the system.” It is estimated that this adjustment wiil save US$ 1 million a year.

An elephant guided by a small wand

With its audacity of heading into territories previously avoided by science and its inherent appeal towards interdisciplinary investigations, the theory of chaos has the smell of something new, of the latest generation. It is true that research in the area only took off during the decade of the 80s, but few people know that this theory began to take shape more than 100 years ago. The forerunner of chaotic dynamics was the French mathematician Henri Poincaré (1854-1912), a nearsighted man, left-handed and clumsy, who, among other his accomplishments, formulated before Einstein a special theory of relativity.

“He was involved with the study of the trajectory of the earth, with the famous problem of the three bodies (Sun, Earth and the Moon) in gravitational attraction, which had challenged scientists since the time of Newton. On investigating this theme in depth, Poincaré ended up discovering the conceptual principles of the theory of chaos”, points out Caldas. Poincaré lived in the Paris of the Belle Époque. Afterwards, research into chaos was practically abandoned due to the enormous difficulties of the calculations that it involved.With the onset of computers, the question returned.

“The initial tendency of the scientists was to eliminate from their equations the non-linear terms” sums up Grebogi. “We wanted that a physical system or a machine would behave in a periodic and predictable manner. However, afterwards we perceived the advantages of chaos. It gives flexibility to the system. Thanks to chaos, it is possible to control very large systems by way of very small disturbances. It is like guiding an elephant with a wand, by hitting in the right place.”

The Project
Non-linear Dynamics; Modality Thematic project; Coordinator Dr. Iberê Luis Caldas – Physics Institute of USP; Investment R$ 99,120.00 and US$ 61,852.00