 TOPICS  # Sharkovsky's Theorem

Order the natural numbers as follows: Now let be a continuous function from the reals to the reals and suppose in the above ordering. Then if has a point of least period , then also has a point of least period .

A special case of this general result, also known as Sharkovsky's theorem, states that if a continuous real function has a periodic point with period 3, then there is a periodic point of period for every integer .

A converse to Sharkovsky's theorem says that if in the above ordering, then we can find a continuous function which has a point of least period , but does not have any points of least period (Elaydi 1996). For example, there is a continuous function with no points of least period 3 but having points of all other least periods.

Sharkovsky's theorem includes the period three theorem as a special case (Borwein and Bailey 2003, p. 79).

Least Period, Mann Iteration, Period Three Theorem

## Explore with Wolfram|Alpha More things to try:

## References

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Conway, J. H. and Guy, R. K. "Periodic Points." In The Book of Numbers. New York: Springer-Verlag, pp. 207-208, 1996.Devaney, R. L. An Introduction to Chaotic Dynamical Systems, 2nd ed. Reading, MA: Addison-Wesley, 1989.Elaydi, S. "On a Converse of Sharkovsky's Theorem." Amer. Math. Monthly 103, 386-392, 1996.Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 49, 1993.S_arkov'skiĭ, O. M. "Co-Existence of Cycles of a Continuous Mapping of a Line onto Itself." Ukranian Math. Z. 16, 61-71, 1964.Stefan, P. "A Theorem of Sharkovsky on the Existence of Periodic Orbits of Continuous Endomorphisms of the Real Line." Comm. Math. Phys. 54, 237-248, 1977.

## Referenced on Wolfram|Alpha

Sharkovsky's Theorem

## Cite this as:

Weisstein, Eric W. "Sharkovsky's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SharkovskysTheorem.html