Discrete dynamical systems — fixed points, periodic orbits, basins

Parent: orbit

The base layer: $X$ a set (for stability questions, a metric space), $f: X \to X$, orbits $O_f(x)$ as in orbit.

The vocabulary

Definition

• $x^*$ is a fixed point if $f(x^*) = x^*$.
• $x$ is a periodic point of period $p$ if $f^p(x) = x$ and $p$ is minimal with this property; its orbit $\{x, f(x), \dots, f^{p-1}(x)\}$ is a $p$-cycle.
• $x$ is eventually periodic if $f^m(x)$ is periodic for some $m \ge 0$ (on a finite $X$, every point is eventually periodic — pigeonhole).
• For $X$ metric: a fixed point $x^*$ is attracting if it has a neighborhood $U$ with $f(U) \subseteq U$ and $f^n(u) \to x^*$ for all $u \in U$. The basin of attraction is $\{x : f^n(x) \to x^*\}$.

Stability test (one dimension)

Let $f: \mathbb{R} \to \mathbb{R}$ be $C^1$ with $f(x^*) = x^*$. If $|f'(x^*)| < 1$ then $x^*$ is attracting; if $|f'(x^*)| > 1$, repelling. For a $p$-cycle $\{x_1,\dots,x_p\}$ apply this to $f^p$; by the chain rule $(f^p)'(x_1) = f'(x_1) f'(x_2) \cdots f'(x_p)$ — one number decides the whole cycle.

Conjugacy is the notion of "same system": $g = h \circ f \circ h^{-1}$ for a bijection $h$ (homeomorphism, in the topological category). Conjugacy maps orbits to orbits, periods to periods — it is how questions get transported between systems (used constantly in symbolic-dynamics and complex-dynamics).

Worked example: the logistic map

$f_r(x) = r x (1-x)$ on $[0,1]$, parameter $0 < r \le 4$. Then $f_r'(x) = r - 2rx$.

Fixed points: solve $rx(1-x) = x$: either $x^* = 0$ or $x^* = 1 - \tfrac{1}{r}$.

• At $0$: $f_r'(0) = r$, attracting iff $r < 1$.
• At $1 - \tfrac{1}{r}$: $f_r'\!\left(1-\tfrac{1}{r}\right) = r - 2r\left(1 - \tfrac{1}{r}\right) = 2 - r$, attracting iff $|2 - r| < 1$, i.e. $1 < r < 3$.

At $r = 3$ the fixed point loses stability and a 2-cycle is born. Count along at $r = 3.2$: the 2-cycle points solve $f(f(x)) = x$ with the fixed points factored out; numerically $x_1 \approx 0.5130$, $x_2 \approx 0.7995$. Check they swap:

Stability of the cycle: $(f^2)'(x_1) = f'(x_1)\,f'(x_2) = (3.2 - 6.4\cdot 0.5130)(3.2 - 6.4\cdot 0.7995) \approx (-0.083)(-1.917) \approx 0.16$, and $|0.16| < 1$: attracting. ✓

Raising $r$ further doubles the period again and again (the period-doubling cascade), reaching chaos at $r \approx 3.5699$; at $r = 4$ the map is conjugate to the tent map and fully chaotic — the interval story continues in topological-dynamics, its statistics in ergodic-theory-of-orbits.