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Mathematics

Creativity in chaos

An award conceded to a Brazilian researcher reveals successes in the building of an innovative theory about dynamic systems

ROBBY A. BERRY / GERHARD WESPThe recent award to the 45 year old Portuguese-Brazilian Marcelo Viana, provided evidence of the research excellence carried out in the country in the area of dynamic systems, the mathematical discipline that studies the types of phenomena that evolve over time, such as climate, chemical reactions, planetary systems and ecological environments. Viana was awarded the University of Coimbra Award, attributed annually to personalities with innovative contributions in the areas of culture and science. A researcher and assistant director of the Pure and Applied Mathematics Institute (IMPA in the Portuguese acronym), of Rio de Janeiro, Viana represents a group of mathematicians in activity in the country who are searching to formulate an all encompassing approach about the general behavior of dynamic systems. And they can count upon a summation of good results. “The theory is under construction. There was already a previous attempt during the decade of the 1960’s, but now we have an innovative point of view, which we’re attempting to prove. This is a long term collective project”, says Viana.

Dynamic systems are a relatively new area of mathematics, of around one hundred years. It came about with the ambition to resolve problems linked to astronomy and celestial mechanics, in an attempt to evaluate the future behavior of the solar system planets and to anticipate if they were going to crash into each other. It so happens that over the last few decades a growing number of phenomena have gone on to be seen as a complex dynamic system. “Interdisciplinary groups have emerged for the study of dynamic systems in various areas of knowledge and both mathematicians and physicists are participating in important collaborations”, says Carmen Prado, a professor at the Physics Institute of the University of São Paulo (USP) and one of the authors of the book entitled, Caos, uma introdução [Chaos, an introduction] (Publisher: Edgard Blücher, 1994).

A classical application of these systems gives one an understanding of what may happen when two animal or vegetable species compete. On can take as an example an environment shared by foxes and hares. If the number of foxes doubles, what will happen with their prey? If the system were to function in a linear fashion, the hares would disappear. It so happens that one is dealing with a dynamic and not a linear system. The result can be the inverse of that expected – in the first moments the hares die off, but, in the medium term, there is going to be lack of food for the foxes and they can starve to death, leaving the ground open for the few hares remaining to repopulate the environment. Another possibility is the approach of cycles in which now one species advances, and then the other dominates. The equations of dynamic systems aim to forecast the most probable destiny or destinies, in conception called attractors.

Butterfly effect
In the case of the dynamic systems called chaotic, the difficulty in making forecasts is much greater and inherent to the mathematical properties of the equations that describe its dynamism. One of the main problems consists in establishing variables with exactitude. Since no magnitude can be measured with infinite precision, any small mistake amplifies itself over time and turns the forecast wrong. This phenomenon produces what has been conventionally called the butterfly effect: a small initial disturbance (a butterfly beating its wings in the Southern Hemisphere) multiplies and accumulates; completely altering the final result (it influences the occurrence of a storm in China).

The work of the IMPA group aims to understand the evolution of dynamic systems making probability forecasts and evaluate if this chance of error is important or negligible. “Making the comparison a little simplistic, we never know if a coin tossed in the air will come down heads or tails. But we can state with absolute certainty that, if we were to toss the coin up 1 million times, from 49% to 51% would be heads and the rest would be tails”, explains  Marcelo Viana.

Born in Rio de Janeiro, the son of Portuguese parents, Marcelo graduated in mathematics from the University of Porto. On his return to Brazil, he wrote his doctorate thesis at IMPA and carried out post-doctorate work at the universities of Princeton and California, in the United States. Among his main contributions, it is possible to highlight the article published in 2005 in the magazine Acta Matematica, in partnership with Artur Avila, a researcher at IMPA/CNRS, proving a conjecture that had been proposed at the start of the decade of the 1990’s by the Russian mathematicians Anton Zorich and Maxim Kontsevich. He has carried out research, including his doctorate thesis, around the so called strange attractors, geometric zones of extravagant forms described in the decade of the 1960’s by the American meteorologist Edward Lorenz, which illustrate the unfolding of chaotic systems. One of these articles published in 1999 in the magazine Inventiones Mathematicae in partnership with the Swedish mathematician Michael Benedicks, gave a solution to a problem presented by the Russian Yakov Sinai and the Frenchman David Ruelle about the type of strange attractor that had been proposed by the French astronomer Michel Hénon.

However, in order to understand the dimension of Viana’s work, one needs to put it into context in the effort carried out by IMPA, whose scientific standard and intellectual environment is comparable to the best institutions in the world. “The institute is an obligatory reference in dynamic systems. Every researcher in this area knows the importance of the IMPA”, says Eduardo Colli, a professor at USP’s Mathematics and Statistics Institute (IME). The seed was planted during the decade of the 1960’s by the mathematician Maurício Peixoto, today 86 years of age, and a pioneer in dynamic systems in the country. Peixoto established a partnership with the American Stephen Smale, from Michigan University, a great scholar on the subject, which until today renders fruit in Brazil.

Smale arrived here and spent six months working at the IMPA and became the supervisor of outstanding names in Brazilian mathematics such as the award winner Jacob Palis Jr. and César Camacho, IMPA’s current director. This second generation of researchers initiated their work during the decade of the 1990’s in an attempt to formulate a global coverage of the behavior of dynamic systems. Various conjectures were formulated by  Palis and the Institute’s mathematicians inclined themselves towards them with the ambition of proving them. Advances have been made on various fronts by mathematician Viana, who had Palis as his supervisor during his doctorate thesis, is an example of this. But, as the researcher himself said, one is dealing with a collective project.

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