⚽ Goals: Look closer at chaotic attractors to understand

• What causes SIC
• how a differential equation can generate a fractal attractor

📖 Definition: Strange attractors have 2 properties:

• The trajectories remain confided to a bounded region of phase space
• Yet, they separate from their neighbours exponentially fast

Different maps

🗒️ Note: the names of the map are not necessarily official and come from Strogatz

Pastry map

Strange attractors arise when

• the flow contracts in some directions (dissipation
• The flow stretches in other directions (SIC)
• The distorted blob must be folded back onto itself to remain a bound region

Comparing the cross section of $S_1$ with $S_2$ and $S_3$ we see it looks like the Cantor set

In fact at $n\to \infin$ it becomes the Cantor set

💼 Case: Setup

Iterations:

Baker’s map

The Backer’s map $B$ if the square $x,y\in[0,1]$ is

$$ (x_{n+1},y_{n+1})=\left \{ \begin{matrix} \begin{pmatrix} 2x_n &ay_n \end{pmatrix} & \text{for} & 0\le x_n \le \frac 12 \\ \begin{pmatrix} 2x_n-1 &ay_n+\frac 12 \end{pmatrix} & \text{for} & \frac 12 \le x_n \le 1

\end{matrix} \right . $$

where $a\in (0,\frac 12]$

Process:

1. The square is stretched and flattened into a $2\times a$ rectangle
2. The rectangle is cut in half yielding two $1\times a$ rectangles and the right half is stacked on top of the left such that its base is $y=\frac 12$

🗒️ Note: the gaps have width $\frac 12 -a$

If we take it to $n\to \infin$ we get a Cantor set though its a more general Cantor set as the spacing are not perfectly $1/3$

Properties of baker map

• Chaotic due $x$-direction stretching
• The fractal set constructed by iterating the map is called the chaotic attractor of the map
• Fractal attractor is a strange attractor

Hénon Map

$$ \begin{pmatrix} x_{n+1} \\ y_{n+1} \end{pmatrix}=\begin{pmatrix} y_n+1-ax^2_n \\ bx_n \end{pmatrix} $$

where $a$ and $b$ are adjustable parameters where $b\in (-1,1)$

Construction: