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Concept
Sharkovskii's theorem
Summary
In mathematics, Sharkovskii's theorem (also spelled Sharkovsky's theorem, Sharkovskiy's theorem, Šarkovskii's theorem or Sarkovskii's theorem), named after Oleksandr Mykolayovych Sharkovsky, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.
For some interval , suppose that
is a continuous function. The number is called a periodic point of period if , where denotes the iterated function obtained by composition of copies of . The number is said to have least period if, in addition, for all . Sharkovskii's theorem concerns the possible least periods of periodic points of . Consider the following ordering of the positive integers, sometimes called the Sharkovskii ordering:
It consists of:
the odd numbers in increasing order,
2 times the odd numbers in increasing order,
4 times the odd numbers in increasing order,
8 times the odd numbers ,
etc.
finally, the powers of two in decreasing order.
This ordering is a total order: every positive integer appears exactly once somewhere on this list. However, it is not a well-order. In a well-order, every subset would have an earliest element, but in this order there is no earliest power of two.
Sharkovskii's theorem states that if has a periodic point of least period , and precedes in the above ordering, then has also a periodic point of least period .
One consequence is that if has only finitely many periodic points, then they must all have periods that are powers of two. Furthermore, if there is a periodic point of period three, then there are periodic points of all other periods.
Sharkovskii's theorem does not state that there are stable cycles of those periods, just that there are cycles of those periods. For systems such as the logistic map, the bifurcation diagram shows a range of parameter values for which apparently the only cycle has period 3.
Official source
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