We all know how a simple action at a critical moment can change our lives. Over the past half-century, with the growing evidence of how small changes can lead to dramatic developments, there has been a paradigm shift in science. Earlier attempts to predict the future as if it were determined with certainty have given way to a more probabilistic approach where uncertainty reigns [TM251 or search for “thatsmaths” at irishtimes.com].
It was a meteorologist, Edward Lorenz, who first drew attention to the sensitivity of the atmosphere to small perturbations. He encapsulated the problem in what is now known as the butterfly effect, asking if the flap of a butterfly’s wing in Brazil can set off a chain of events that later triggers the formation of a tornado in Texas.
Lorenz demonstrated, with skill,
The chaos of heatwave and chill:
Tornadoes in Texas
Are formed by the flexes
Of butterflies’ wings in Brazil.
If slightly altered initial conditions can lead to greatly differing solutions, long-range prediction may be impossible. Even if we have a perfect model of the atmosphere, the fluttering of a butterfly’s wings could alter the initial conditions and thus change the long-term prediction drastically. This is what scientists now mean by chaotic behaviour.
Weather prediction has been powerfully influenced by this change of perspective. Instead of asking if it will rain next Thursday, we ask for the likelihood of various levels of precipitation. This is answered by performing a suite of forecasts and analysing the statistics of the ensemble of solutions.
Sharkovsky’s Theorem
Oleksandr Sharkovsky (1936-2022) was a Ukrainian mathematician, renowned for proving a theorem on periodic solutions of discrete dynamical systems. His mathematical career started impressively when, aged just 15, he was the winner of the Kyiv Mathematical Olympiad for schoolchildren. He published a scientific paper in his first year at Kyiv National University. He won many prestigious prizes and awards for his work. Sharkovsky died just one year ago, on 21 November 2022, at the age of 85.
In 1964, Sharkovsky proved a remarkable theorem, which is now of central relevance in dynamic systems theory. But his discovery attracted little notice at first. The subject area was not in vogue and his work, published in Russian, was not available in translation.
In 1975, Tien-Yien Li and James Yorke published a paper entitled “Period three implies chaos”, in which they proved that in a simple system, if there is a solution with period 3 (repeating itself every 3 clock-ticks), then it has solutions with every other period. This is a special case of the theorem proved earlier by Sharkovsky, although Li and Yorke were unaware of his work. They dubbed this behaviour “chaotic”, introducing the term into the scientific lexicon and sparking the rapid development of chaos theory. Chaotic behaviour, although determined by an equation, is indistinguishable from a completely random process.
Attending a conference in East Berlin, Yorke met Sharkovsky, who told him about the result now known as Sharkovsky’s Theorem. So, Li and Yorke’s paper led to global recognition of Sharkovsky’s work. The Sharkovsky theorem is now a standard mathematical result and is amongst the most original theorems of the 20th century.
Trailer
Next week, That’s Maths will post an article about the Sharkovsky Ordering.
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