Sarkovskii theorem

The sharkovskii's theorem is a set of mathematics , of an important statement about the possible periods in the iteration of a continuous function makes. A special case of the theorem is the statement that a continuous dynamic system on the real line with a point of order 3 already has points for every order. This is often phrased briefly to say that Period 3 implies chaos .

The sentence

Be

{\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}

a continuous function. It is said that a periodic point of order (or the period length) m is when (where the fold link of referred to itself) and for all . The statement is about the possible orders of periodic points of . To formulate it, consider the so-called Sarkovskii order of natural numbers . It is about total order ${\ displaystyle x}$ ${\ displaystyle f ^ {m} (x) = x}$ ${\ displaystyle f ^ {m}}$ ${\ displaystyle m}$ ${\ displaystyle f}$ ${\ displaystyle f ^ {k} (x) \ not = x}$ ${\ displaystyle 0 <k <m}$ ${\ displaystyle f}$

{\ displaystyle 3,5,7,9, \ ldots, 2 \ cdot 3.2 \ cdot 5.2 \ cdot 7, \ ldots, 2 ^ {2} \ cdot 3.2 ^ {2} \ cdot 5, ....., 2 ^ {4}, 2 ^ {3}, 2 ^ {2}, 2,1.}

This order starts with the odd numbers in ascending order, followed by twice the odd numbers, four times the odd numbers, etc., and ends with the powers of two in descending order.

The Sarkovskii theorem says that if it has a periodic point of length and in the Sarkovskii order it holds that there is then also (at least) one periodic point of length . ${\ displaystyle f}$ ${\ displaystyle m}$ ${\ displaystyle m \ leq n}$ ${\ displaystyle n}$

Conclusions and remarks

The sentence has several consequences. On the one hand, if only has finitely many periodic points, then they must all have a power of two as order. If it has any periodic point, it also has a fixed point . Furthermore, as soon as there is a point of order , there are periodic points for every order. This statement is also called the Li and Yorke theorem. ${\ displaystyle f}$ ${\ displaystyle f}$ ${\ displaystyle f}$ ${\ displaystyle 3}$

Sarkovski's theorem is optimal in the sense that for every natural number a continuous function can be constructed in such a way that for every natural number that comes in the Sarkovskii order (inclusive ) there are periodic points with this period length, but none periodic points of smaller order. For example, there are functions that have no periodic points of length , but do have for all other numbers (period does not imply chaos). ${\ displaystyle m}$ ${\ displaystyle m}$ ${\ displaystyle m}$ ${\ displaystyle 3}$ ${\ displaystyle 5}$

Sarkovski's theorem does not apply to dynamic systems on other topological spaces . For the rotation of the circular line by 120 degrees (third turn ), each point is periodic with the length , and no further period lengths appear. ${\ displaystyle 3}$

history

This theorem was proven by the Ukrainian mathematician Oleksandr Scharkowskyj in 1964 and remained unnoticed for a long time. Around 10 years later, Li and Yorke proved the special case that period implies chaos, without knowing the original result . ${\ displaystyle 3}$

literature

John H. Argyris: The Exploration of Chaos . An introduction to the theory of nonlinear systems. Completely revised and expanded 2nd edition. Springer, Berlin / Heidelberg 2010, ISBN 978-3-540-71071-4 .
Wolfgang Metzler: Nonlinear Dynamics and Chaos: An Introduction . Teubner Study Books: Mathematics. Teubner, Stuttgart / Leipzig 1998, ISBN 3-519-02391-1 , 4th chapter, p. 45 ff .