Asymptotically Periodic

Let $f$ be a smooth map on $\mathbb{R}$. An orbit $\{x_1,x_2,\dots \}$ is called asymptotically periodic if it converges to a periodic orbit as $n\to \infty$. In other words, there exists a periodic orbit $\{y_1,y_2,\dots ,y_k,y_1,\dots \}$ such that:

$$\underset{n\to \infty }{\lim }|x_n-y_n|=0$$

(Eventually Periodic orbits are AP — they hit the cycle exactly in finite time, which is a special case. The converse doesn’t hold.)

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The typical AP orbit: $x_1$ in the basin of a period-$k$ sink $\{p_1,\dots ,p_k\}$. The orbit spirals toward the cycle with Lyapunov exponent:

$$h(x_1)=\frac{1}{k}\left(\ln |f^{\prime }(p_1)|+\cdots +\ln |f^{\prime }(p_k)|\right)<0$$

since the sink condition forces the product of derivatives to be less than 1.

Chaotic orbits are defined in part by not being AP — bounded but never settling into any periodic behavior.