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.^[1] 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.

Statement

For some interval ${\displaystyle I\subset \mathbb {R} }$, suppose that

is a continuous function. The number ${\displaystyle x}$ is called a periodic point of period ${\displaystyle m}$ if ${\displaystyle f^{(m)}(x)=x}$, where ${\displaystyle f^{(m)}}$ denotes the iterated function obtained by composition of ${\displaystyle m}$ copies of ${\displaystyle f}$. The number ${\displaystyle x}$ is said to have least period ${\displaystyle m}$ if, in addition, ${\displaystyle f^{(k)}(x)\neq x}$ for all ${\displaystyle 0<k<m}$. Sharkovskii's theorem concerns the possible least periods of periodic points of ${\displaystyle f}$. Consider the following ordering of the positive integers, sometimes called the Sharkovskii ordering:^[2]

It consists of:

• the odd numbers ${\displaystyle =(2n+1)\cdot 2^{0}}$ in increasing order,
• 2 times the odd numbers ${\displaystyle =(2n+1)\cdot 2^{1}}$ in increasing order,
• 4 times the odd numbers ${\displaystyle =(2n+1)\cdot 2^{2}}$ in increasing order,
• 8 times the odd numbers ${\displaystyle =(2n+1)\cdot 2^{3}}$,
• etc. ${\displaystyle =(2n+1)\cdot 2^{m}}$
• finally, the powers of two ${\displaystyle =2^{n}}$ 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 ${\displaystyle f}$ has a periodic point of least period ${\displaystyle m}$, and ${\displaystyle m}$ precedes ${\displaystyle n}$ in the above ordering, then ${\displaystyle f}$ has also a periodic point of least period ${\displaystyle n}$.

One consequence is that if ${\displaystyle f}$ 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. In fact, there must be cycles of all periods there, but they are not stable and therefore not visible on the computer-generated picture.

The assumption of continuity is important. Without this assumption, the discontinuous piecewise linear function ${\displaystyle f:0,3)\to 0,3)}$ defined as:

for which every value has period 3, would be a counterexample. Similarly essential is the assumption of ${\displaystyle f}$ being defined on an interval. Otherwise ${\displaystyle f:x\mapsto (1-x)^{-1}}$, which is defined on real numbers except the one: ${\displaystyle \mathbb {R} \setminus \{1\},}$ and for which every non-zero value has period 3, would be a counterexample.

Generalizations and related resultsedit

Sharkovskii also proved the converse theorem: every upper set of the above order is the set of periods for some continuous function from an interval to itself. In fact all such sets of periods are achieved by the family of functions ${\displaystyle T_{h}:0,1\to 0,1}$, $$Zdroj:https://en.wikipedia.org?pojem=Sharkovskii_theorem
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