--- author: - "[Eugeniy E. Mikhailov](http://physics.wm.edu/~evmik/) and [Greg Bentsen](https://physics.wm.edu/~gbentsen/)" title: Iterated maps --- ## Agenda Today - Iterated maps, fractals, and chaos \ **Big idea:**  Simple deterministic rules can generate surprisingly complicated behavior when repeated over and over. \ Three examples: - Newton-Raphson root-finding as a dynamical system - Mandelbrot Set and fractals - Logistic Map and population growth # Iterated Maps ## Iterated maps An **iterated map** repeatedly applies the same rule: $$x_{n+1} = F(x_n).$$ Starting from an initial value $x_0$, we generate an **orbit** $$x_0,\quad x_1,\quad x_2,\quad x_3,\quad \ldots$$ Computationally: ``` {.python .numberLines} x = x0 for n in range(N): x = F(x) ``` \ Today we will ask: **What happens after we apply $F$ over and over again?** ## A simple linear example Consider the iterated linear map $$ F(x)=a x. $$ Starting from $x_0$, the orbit is $x_0, \quad a x_0, \quad a^2 x_0, \quad \ldots, \quad$ The exact solution is therefore $$ x_n=a^n x_0. $$ Different values of $a$ give qualitatively different long-time behavior. Starting from $x_0=1$: ::: columns ::: {.column width="45%"} $a=\frac{1}{2}$ convergence to $0$ $$ 1, \quad \frac12, \quad \frac14, \quad \frac18, \quad \cdots, \quad 0 $$ \ $a=-\frac{1}{2}$ oscillatory convergence to $0$ $$ 1, \quad -\frac12, \quad \frac14, \quad -\frac18, \quad \cdots, \quad 0 $$ ::: ::: {.column width="45%"} $a=-1$ persistent oscillation $$ 1, \quad -1, \quad 1, \quad -1, \quad \cdots $$ \ \ $a=2$ divergence $$ 1, \quad 2, \quad 4, \quad 8, \quad \cdots $$ ::: ::: ## Fixed points A **fixed point** $x^*$ satisfies $$F(x^*) = x^*$$ meaning that the orbit is **constant for all time**. - For $F(x) = a x$ the fixed point is $x^* = 0$. - For $F(x) = x^2$ there are two fixed points: $x^* = 0,1$. \ If the orbit approaches $x^*$, $$x_n \rightarrow x^*,$$ then $x^*$ is a **stable** (attractive) fixed point. If the orbit diverges from the fixed point, it is **repulsive**. ## Stability of a fixed point Suppose we start close to a fixed point: $$x_n = x^* + \delta_n.$$ For small $\delta_n$ we may Taylor expand: $$ \begin{aligned} F(x^*+\delta_n) &\approx F(x^*) + F'(x^*)\delta_n \\ &= x^* + \delta_{n+1} \end{aligned} $$ Since $F(x^*)=x^*$, $$\delta_{n+1} \approx F'(x^*)\delta_n.$$ Therefore: $$|F'(x^*)| < 1 \quad \Rightarrow \quad \text{stable}$$ $$|F'(x^*)| > 1 \quad \Rightarrow \quad \text{unstable}$$ \ ## Stability in the linear example $$ F(x)=a x. $$ Since $$ F'(x)=a, $$ the fixed point $x^*=0$ is stable when $$ |a|<1, \quad \quad \quad \quad \quad \quad \quad a = \frac{1}{2}: \quad \quad 1, \quad \frac12, \quad \frac14, \quad \frac18, \quad \cdots, \quad 0 $$ and unstable when $$ |a|>1, \quad \quad \quad \quad \quad \quad a = 2: \quad \quad 1, \quad 2, \quad 4, \quad 8, \quad \cdots $$ and marginal when $$ |a|=1, \quad \quad \quad \quad \quad \quad a = -1: \quad \quad 1, \quad -1, \quad 1, \quad -1, \quad \cdots $$ ## A nonlinear example For the linear example, it doesn't matter what initial condition $x_0$ you choose. The iterated map is **globally** stable or unstable depending on the parameter $a$. Things get more interesting for **nonlinear** maps. Consider $$F(x) = x^2$$ The fixed points $x^*$ satisfy $x^* = (x^*)^2$, so we have two fixed points $$x^* = 0, \quad 1$$ Are these fixed points stable or unstable? Use the derivative criterion $|F'(x)| < 1$. $$F'(x) = 2 x$$ - At $x^* = 0$ we have a **stable** fixed point since $|F'(0)| = 0$. - At $x^* = 1$ we have an **unstable** fixed point since $|F'(1)| = 2$. ## Cobweb plots We can find and study fixed points visually in the $x-y$ plane. To do this, let's split each step of the iterated map onto two pieces: $$y = F(x) \quad \quad \quad \quad x = y$$ Fixed points satisfy $F(x^*) = x^*$. This can be viewed as the **intersection** of these two functions. And long-term behavior can be viewed as "ping-ponging" back and forth between these two functions. ::: columns ::: {.column width="45%"} ::: {style="text-align: center;"} ![$F(x) = a x, a = 1/2$](./pics/cobweb_linear.png){width="80%"} ::: ::: ::: {.column width="45%"} ::: {style="text-align: center;"} ![$F(x) = x^2$](./pics/cobweb_quad.png){width="80%"} ::: ::: ::: # Newton-Raphson as an Iterated Map ## Newton-Raphson revisited Newton-Raphson finds roots of $$f(x)=0$$ using $$x_{n+1} = x_n-\frac{f(x_n)}{f'(x_n)}.$$ Define the **Newton map** $$F(x)=x-\frac{f(x)}{f'(x)}.$$ Then Newton-Raphson is simply $$x_{n+1}=F(x_n).$$ \ Every root $x^*$ of $f$ is a fixed point: $$f(x^*)=0 \quad\Rightarrow\quad F(x^*)=x^*.$$ ## Root-finding in the complex plane Consider finding roots of $$f(z)=z^3-1.$$ The roots are **complex** $$z_1=1,$$ $$z_2=e^{2\pi i/3}, \qquad z_3=e^{4\pi i/3}.$$ Q: can we find these roots via Newton-Raphson? From the previous slide we obtain the Newton map: $$ F(z) = z-\frac{f(z)}{f'(z)} = z-\frac{z^3-1}{3z^2}$$ or $$ F(z) = \frac{2}{3}z+\frac{1}{3z^2}$$ Clearly the roots $z_1,z_2,z_3$ are fixed points of this map.\ **Question:**  Given an initial guess $z_0$, which root does Newton-Raphson find? ## Demo: Newton-Raphson in the complex plane For each starting point $$z_0=x+iy,$$ run Newton-Raphson until it converges. Color the starting point according to the root it reaches. \ Algorithm: 1. Make a grid of initial conditions $z_0$. 2. Iterate Newton-Raphson at every point. 3. Determine which root each point approaches. 4. Assign each root a different color. ## Basins of attraction Each root has a **basin of attraction**: $$B_i = \left\{ z_0: z_n \rightarrow z_i \right\}.$$ Points in the same basin converge to the same root. ::: {style="text-align: center;"} ![Source: https://paulbourke.net/fractals/newtonraphson/](./pics/newton_fractal.jpg){width="55%"} ::: ## What happens near the boundary? Points that begin very close together can converge to **different roots**. Zooming into the boundary reveals new structure at smaller and smaller scales. \ This is our first example of a **fractal**. A useful working definition: > A fractal has nontrivial structure that persists as we examine it at > finer scales. ## A lesson from Newton-Raphson Newton-Raphson is: - Deterministic - Defined by a simple formula $F(z) = \frac{2}{3}z+\frac{1}{3z^2}$ - Easy to implement Yet its global behavior can be extremely complicated. \ The complexity does **not** come from randomness. It comes from repeatedly applying a nonlinear map. \ **Simple rule + iteration $\rightarrow$ complex structure** # Mandelbrot Set ## An even simpler map Consider $$\boxed{ z_{n+1}=z_n^2+c }$$ with $$z_0=0$$ and $c = a+i b$ a complex number. For each complex value of $c$, generate the orbit $$0,\quad c,\quad c^2+c,\quad \ldots$$ Then ask: **Does the orbit remain bounded, or does it escape to infinity?** ## Demo: iterate $z_{n+1}=z_n^2+c$ For one value of $c$: ``` {.python .numberLines} z = 0 for n in range(N): z = z*z + c if abs(z) > 2: break ``` Try several choices of $c$. Observe: - Some orbits escape rapidly. - Some settle into simple behavior. - Some remain bounded but behave in complicated ways. ## The Mandelbrot set The **Mandelbrot set** is the collection of parameters $c$ for which the orbit remains bounded: $$ \mathcal{M} = \left\{ c\in\mathbb{C}: z_{n+1}=z_n^2+c,\; z_0=0 \text{ remains bounded} \right\}. $$ ::: {style="text-align: center;"} ![](./pics/mandelbrot.jpeg){width="70%"} ::: ## The Mandelbrot set, fractals, and self-similarity ::: columns ::: {.column width="45%"} ::: {style="text-align: center;"} ![](./pics/mandelbrot_zoom.jpeg){width="80%"} ::: ::: {style="text-align: center;"} ![](./pics/mandelbrot_zoom2.jpeg){width="80%"} ::: ::: ::: {.column width="45%"} ::: {style="text-align: center;"} ![](./pics/mandelbrot_zoom1.jpeg){width="80%"} ::: ::: {style="text-align: center;"} ![](./pics/mandelbrot_zoom3.jpeg){width="80%"} ::: ::: ::: ## Newton fractal vs. Mandelbrot set Newton fractal Mandelbrot set ---------------------------------- -------------------------------- Fix the map Change the map Vary the initial condition $z_0$ Vary the parameter $c$ Ask: **Which attractor?** Ask: **Bounded or unbounded?** \ Both reveal complicated structure by running a simple algorithm many times. \ **Computational strategy:**  Scan over many inputs, iterate, then classify the long-time behavior. # Logistic Map ## Population growth Suppose $x_n$ is the population during generation $n$, normalized so that $$0\leq x_n\leq 1.$$ With unlimited resources, we might expect $$x_{n+1}=r x_n.$$ But unlimited exponential growth is unrealistic. Growth should slow when the population becomes large. ## The logistic map A simple discrete model is $$\boxed{ x_{n+1}=r x_n(1-x_n) }.$$ Interpretation: $$ \underbrace{x_n}_{\text{current population}} \; \underbrace{(1-x_n)}_{\text{limited resources}} \; \underbrace{r}_{\text{growth parameter}} $$ For $0\leq r\leq4$ and suitable initial conditions, the population remains between 0 and 1. \ **Question:**  What happens to the population after many generations? ## Fixed points of the logistic map A fixed population satisfies $$x^*=r x^*(1-x^*).$$ Therefore the two fixed points are $$x^*=0$$ and $$x^*=1-\frac{1}{r}.$$ But finding fixed points is not enough. \ We must also ask: **Is the fixed point stable?** ## Stability changes as $r$ changes ::: columns ::: {.column width="45%"} For $$F(x)=rx(1-x),$$ we have $$F'(x)=r(1-2x).$$ At the **trivial** fixed point $x^* = 0$ we have $F'(0) = r$, so this point is stable for $0 \leq r < 1$. \ At the **nonzero** fixed point $x^* = 1 - 1/r$, $$F'(x^*)=2-r$$ So stability $|F'(x^*)| < 1$ requires $1