We are tasked with proving that for each positive integer $k$, the function $g(x) = 4x(1 - x)$ has an orbit of period $k$. In other words, we need to show that for any $k \in \mathbb{Z}^+$, there exists some point $x_0$ such that iterating $g(x)$ on $x_0$ produces a repeating cycle of length $k$.

Step 1: Understanding periodic points

An orbit of period $k$ means there is some point $x_0$ such that after $k$ iterations, $g^k(x_0) = x_0$, but for all $1 \leq n < k$, $g^n(x_0) \neq x_0$. Mathematically, this is expressed as: $g^k(x_0) = g(g(\dots g(x_0) \dots)) = x_0$ and $g^n(x_0) \neq x_0 \quad \text{for all} \quad 1 \leq n < k.$ Thus, the function returns to its starting value after exactly $k$ iterations.

Step 2: Behavior of the logistic map

The given function $g(x) = 4x(1 - x)$ is a well-known instance of the logistic map. The logistic map is famous for exhibiting chaotic behavior, including the existence of periodic points and orbits of all possible periods for certain parameter values.

For $g(x) = 4x(1 - x)$, it has been established that this map is chaotic for values like 4, meaning that it indeed exhibits periodic points of all periods.

Step 3: Sharkovskii's Theorem

Sharkovskii's Theorem is a key result in the theory of dynamical systems, particularly in the context of one-dimensional maps like the logistic map. According to this theorem, if a continuous map has a periodic point of period 3, then it must have periodic points of all periods.

The function $g(x) = 4x(1 - x)$ is known to have a periodic orbit of period 3 (this can be explicitly computed or referred to from results on the logistic map). By Sharkovskii's Theorem, this implies that $g(x)$ must also have periodic points for every $k \in \mathbb{Z}^+$.

Step 4: Conclusion

Since $g(x) = 4x(1 - x)$ has periodic points of period 3, and by Sharkovskii’s Theorem this implies periodic points of every period, we conclude that for every positive integer $k$, there is a point $x_0$ such that the orbit of $x_0$ under $g(x)$ has period $k$.

Thus, for every $k \in \mathbb{Z}^+$, there is an orbit of period $k$ for the map $g(x) = 4x(1 - x)$.

Would you like further details on any specific part of the proof or more information on Sharkovskii’s Theorem?

Related Questions:

1. What is Sharkovskii’s Theorem and how does it apply to chaotic systems?
2. How can we explicitly find a periodic orbit of period 3 for $g(x) = 4x(1 - x)$?
3. What is the significance of chaotic behavior in the logistic map?
4. How do bifurcations in the logistic map lead to the emergence of periodic orbits?
5. Can you provide examples of other maps that exhibit chaotic behavior like the logistic map?

Tip:

Sharkovskii's Theorem is a powerful tool in dynamical systems. Learning more about it can give deeper insights into periodic orbits and chaos in one-dimensional maps.