This post is an extension and elaboration oftwo recent posts, with more technical details
Bifurcation diagram for solutions of the logistic equation for . The period-3 band starting at is clear [image from Wikimedia Commons].
“The reasonable man adapts himself to the world:the unreasonable one persists in trying to adapt the world to himself.Therefore all progress depends on the unreasonable man.”
— George Bernard Shaw, Man and Superman.
The reasonable man, if asked to write a list of the natural, or counting, numbers would probably write
written in order of increasing magnitude. The unreasonable man might write the sequence
(the symbol indicates “precedes”). This is the sequence called the Sharkovsky Ordering, and it was first written in a remarkable paper published by Oleksandr Sharkovsky in 1964.
The sequence is constructed as follows: first come the odd numbers, starting with 3, in increasing order. Then follows the same sequence with each odd number multiplied by 2. Next, each odd number multiplied by ; next, multiplied by and so on. Finally, the descending sequence of positive powers of 2, ending with 1. This sequence contains all the positive integers, each occurring just once. It is known as the Sharkovsky Ordering.
Oleksandr Sharkovsky
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. In 1967 he defended his doctoral thesis. He was elected a corresponding member of the Academy of Sciences of the Ukrainian SSR and, in 2006, a full member of the Ukrainian National Academy of Sciences. He is the author of nearly 250 scientific papers and five monographs. 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, Oleksandr Sharkovsky proved a remarkable theorem. If a continuous map of the real line into itself has a periodic cycle of period and if in Sharkovsky’s list, then the map has a cycle of period . This implies that, if there is a cycle of odd period, then cycles of all multiples of 2 exist. Moreover, if there is a period-3 cycle (3 being the first number on the list), then cycles of all periods exist.
With his theorem, Sharkovsky laid the foundations of the topological theory of one-dimensional dynamic systems, a theory now of central relevance for research in diverse fields of science. But his discovery attracted little notice at first. The subject area of combinatorial dynamics was not in vogue and his work, written in Russian, was not available in translation.
Recognition
Sharkovsky’s work did not become widely known until, in 1975, a famous paper, “Period three implies chaos” by Li and Yorke was published. This included the result that the presence of a periodic point of period 3 implies the presence of periodic points of all other periods. This corresponds to the initial number 3 at the head of the Sharkovsky order. Later, 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.
Sharkovsky’s result is amongst the most original theorems of the 20th century. The theorem gave a strong impulse for new, important, and interesting research and found applications in many fields of science. The Sharkovsky Ordering is almost magical in its power to indicate the order in which cycles of different periods appear.
A Period 3 Cycle in the Logistic Map
The simple logistic map provides an excellent model for illustrating the period-doubling route to chaos and the emergence of period-3 cycle. We posted an article on the logistic equation a few weeks ago [ThatsMaths, 9 Nov 23]. The map is written
and is (for ) a quadratically nonlinear map from the unit interval into itself. As the parameter increases from 2 to 4, the character of the solution changes, assuming a multitude of forms. For , the solution tends to , after oscillating about that value for some time. For , the solution approaches a period 2 cycle, oscillating between two values. For increasing , solutions with successive period doublings, from 4 to 8 to 16 and so on, with the transitions occurring at successively smaller intervals, leading to chaos at the end of the period-doubling cascade. Beyond the point of accumulation () there is an infinite number of fixed points with different periodicities, and an infinite number of different periodic cycles. There are also an uncountable number of initial points which give totally aperiodic trajectories where the pattern never repeats. At , a stable period 3 cycle emerges.
Bifurcation diagram for solutions of the logistic equation for . The period-3 band starting at is clear [image from Wikimedia Commons].Various regimes can be identified in the bifurcation diagram (above). At there is a period-4 cycle and, following this, a cascade with period doubling at each stage. This corresponds to the tail of Sharkovsky’s sequence: . The wide band of regular motion starting at is the period-3 band. Recall that the number 3 stands at the head of the Sharkovsky ordering and, once a period three solution is present, solutions of all other periods are also present.
Applications
There are numerous situations where chaos theory is of value: biological applications include population dynamics, genetics and epidemiology. Examples in economics include commodity quantity and price forecasts, modelling business cycles, and time-series analysis for economic variables. Equations of this sort are also used to simulate communication systems, to analyse the propagation of rumours and disinformation through social media. Also in many other contexts.
Sharkovsky’s Theorem has become one of the classical results of the theory of dynamical systems. The theorem has found applications in many fields of science: physics, chemistry, biology, medicine, economics and elsewhere. The theorem is now a standard mathematical result and his discovery of a remarkable ordering of the natural numbers has made Sharkovsky’s name famous.
Acknowledgement
My thanks to Dr Nigel Buttimore, Fellow Emeritus of Trinity College Dublin, for telling me about Sharkovsky’s Theorem and for providing guidance to sources on it.
Sources
Krzysztof Ciesielski and Zdzislaw Pogoda, 2008: On Ordering the Natural Numbers or the Sharkovski Theorem. Amer. Math. Monthly, 115(2), 159–165.
T.-Y. Li and J. A. Yorke, 1975: Period three implies chaos. Amer. Math. Monthly, 103, 985–992.
Lorenz, E. N., 1963: Deterministic non-periodic flow. J. Atmos. Sci., 20, 130–141.
Sharkovsky, 1964: Coexistence of cycles of a continuous mapping of a line into itself (in Russian). Ukrainian Math. J., 16, 61–71.
