Discrete systems in 1D - Maps

A discrete dynamical system is defined by by an iterative rule

$$ x_{n+1}=f(x_n) $$

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💃 Example: the iterative map of $x_{n+1}=\cos(x_n)$ for all $x_0$

The blue dots are $x_0$ note that they are on a different scale then the rest, they follow $-\pi$ to $\pi$ when the rest goes from $-1$ to $1$

This is a one dimensional map, because line figure, all the points can be represented in one dimensions, here going from -1 to 1 and we see it converges at $x^$ defined $x^=\cos(x^*)$

Orbit map: set of values $x_0,x_1,x_2,\ldots$ starting from an initial value $x_0$ obtain from

$$ x_1=f(x_0) \quad x_2=f(x_1) \quad \ldots \quad x_N=f(x_{N-1}) $$

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Fixed points and cobwebs

💼 Case: $x_{n+1} =f(x_n)$ where $f$ is a smooth function from the real line to itself

📍 Fixed points: $x^$ is a fixed point if $f(x^)=x^*$

We define $\lambda =f'(x^*)$ which allows us to write

⚖️ Stability analysis:

$|\lambda |<1$ fixed point is stable

$|\lambda |>1$ fixed point is unstable

$|\lambda |=1$ is inconclusive higher derivatives need to be considered

🧉 Proof (expand)

Cobweb

🪧 Setup:

Plot both $y=f(x)$ and $y=x$
Write out $x_0$ on the $x$-axis

🪚 Use:

Go from $(x_0,0)$ vertically to your function $f(x)$ (it can be up or down) and the value on the $y$-axis will be $y=f(x_0)=x_1$
Go horizontally from your new point ${(x_0,x_1)}$ to the $x=y$ function giving as your new point $(x_1,x_1)$

Repeat to get to any higher $n$ point $x_n$

🗒️ Note: its basically a fancy lookup table, they are used to see global behaviour at a glance

💃 Example: Logistic map $x_{n+1}=\cos(x_n)$ shown above

Solve to fin $x^*$

$$ \cos(x)=x \quad \Rightarrow \quad x\simeq 0.739 $$
Calculate its derivative

$$ \begin{aligned} (\cos(x^))'&=-\sin(x^)\simeq - 0.674 \\ \Rightarrow \; |\lambda |&=|f'(x^*)|=0.674<1 \end{aligned} $$

Use the stability to say it is stable and if required you can plot the cobweb above

🗒️ Note: if there are multiple $x^*$ you need to analyse each individually

💃 Example: consider the logistic map $x_{n+1}=\frac 12 x^3_n + \frac 12 x_n$

Find the fixed points

$$ \begin{aligned} \frac 12 x^{3}+\frac 12 x^{}&=x^* \\ \Rightarrow \quad x^*&=0,\pm 1 \end{aligned} $$

Calculate the derivative and apply the solutions

$$ f'(x^*)=\frac 32 x^{*2} + \frac 12 \quad \Rightarrow \quad |\lambda_0|=|f'(0)|=\frac 12 <1 \quad |\lambda_{\pm 1}|=|f'(\pm 1)|=2>1 $$