One-Dimensional Dynamics - Main Theorem

TheoremSharkovsky's Theorem

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous map. There exists a total ordering on the positive integers, called the Sharkovsky ordering:

$3 \triangleright 5 \triangleright 7 \triangleright 9 \triangleright \cdots \triangleright 2 \cdot 3 \triangleright 2 \cdot 5 \triangleright \cdots \triangleright 2^2 \cdot 3 \triangleright 2^2 \cdot 5 \triangleright \cdots \triangleright 2^3 \triangleright 2^2 \triangleright 2 \triangleright 1$

If $f$ has a periodic point of period $n$ , and $n \triangleright m$ in the Sharkovsky ordering, then $f$ must have a periodic point of period $m$ .

This remarkable theorem, proven by Oleksandr Sharkovsky in 1964, reveals deep structural constraints on the periods of continuous interval maps. The ordering places odd numbers first (starting with 3), followed by $2$ times odd numbers, then $4$ times odd numbers, and finally powers of $2$ in decreasing order.

The most famous consequence is that if $f$ has a periodic point of period 3, then it must have periodic points of all periods. This result, sometimes called "period three implies chaos," became widely known through the work of Li and Yorke in 1975, though Sharkovsky had discovered the full ordering years earlier.

TheoremSchwarzian Derivative Theorem

Let $f: I \to I$ be a smooth map of an interval with negative Schwarzian derivative everywhere. Then:

$f$ has at most one attracting periodic orbit
The basin of any attracting periodic orbit contains all critical points (where $f'(x) = 0$ )
If $f$ has no attracting periodic orbits, then there exists a unique invariant measure absolutely continuous with respect to Lebesgue measure

This theorem provides strong constraints on the dynamics of maps with $Sf < 0$ , ruling out complicated coexistence of multiple attractors.

The Schwarzian derivative condition is satisfied by many important families, including the logistic map $f_\mu(x) = \mu x(1-x)$ for appropriate parameters. The theorem implies that as parameters vary, the system can have at most one stable periodic orbit at any given time. Transitions between different periodic orbits occur through bifurcations.

Remark

The uniqueness of attracting periodic orbits for negative Schwarzian maps has profound implications for parameter-dependent families. As parameters change continuously, one attracting orbit must disappear before another can appear. This leads to the period-doubling route to chaos observed in many physical systems, where a stable fixed point becomes unstable and is replaced by a stable period-2 orbit, which in turn doubles to period 4, and so on.

ExampleApplication to Logistic Map

For the logistic map $f_\mu(x) = \mu x(1-x)$ , we can verify:

$Sf_\mu(x) = -\frac{6}{x^2(1-x)^2} < 0$

on $(0,1)$ . Therefore, by the Schwarzian derivative theorem:

For $\mu \in (1, 3)$ , the unique attracting fixed point $x^* = 1 - 1/\mu$ attracts the critical point $x = 1/2$
For $\mu \in (3, 1+\sqrt{6})$ , there exists a unique attracting period-2 orbit
As $\mu$ increases, period-doubling bifurcations occur, each creating a unique new attracting periodic orbit

These theorems demonstrate that even in one dimension, continuous maps exhibit rich and complex dynamics. Sharkovsky's theorem reveals universal constraints on periodic orbits, while the Schwarzian derivative theorem explains why many smooth systems undergo predictable bifurcation sequences. Together, they provide a theoretical foundation for understanding the transition from order to chaos in one-dimensional systems.