The Henon map is a two-dimensional generalization of the logistic map:

$x_{n+1} = 1 - ax_n^2 + y_n, \quad y_{n+1} = bx_n$

For the classical parameters $a = 1.4$, $b = 0.3$, the map exhibits a strange attractor with:

• Fractal dimension $d \approx 1.26$
• Positive Lyapunov exponent $\lambda_1 \approx 0.42$
• Cantor-set structure in cross-sections
• Invertibility (since $\det J = -b \neq 0$)

The Henon map demonstrates that discrete two-dimensional maps can be chaotic, unlike continuous two-dimensional flows (by Poincare-Bendixson).

The Arnold cat map on the torus $T^2 = \mathbb{R}^2/\mathbb{Z}^2$ is defined by the matrix action:

$\begin{pmatrix} x_{n+1} \\ y_{n+1} \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} x_n \\ y_n \end{pmatrix} \pmod{1}$

This map:

• Preserves area (det = 1, symplectic)
• Has eigenvalues $\lambda = (3 \pm \sqrt{5})/2$ (golden ratio-related)
• Exhibits hyperbolic chaos: exponential divergence along unstable direction, convergence along stable
• Is ergodic and mixing on $T^2$

Named for Arnold's illustration using a cat's image, which becomes unrecognizable after iterations but eventually recurs (Poincare recurrence).

The Mandelbrot set $M \subset \mathbb{C}$ parameterizes the family $f_c(z) = z^2 + c$:

$M = \{c \in \mathbb{C} : f_c^n(0) \not\to \infty\}$

This set encodes the bifurcation structure of quadratic maps:

• Connected components correspond to different attracting cycles
• The main cardioid: attracting fixed points
• Period-2 bulb: attracting period-2 orbits
• Smaller bulbs: higher periods following the Farey tree structure
• Fractal boundary with infinite detail at all scales

The Mandelbrot set visualizes parameter space structure, showing where different dynamical behaviors occur.

The baker's map (mentioned earlier) exemplifies chaotic mixing: it stretches, cuts, and restacks like kneading dough. After $n$ iterations:

• Horizontal coordinates: $x_n = 2^n x_0 \pmod{1}$ (binary shift)
• Vertical coordinates: compressed by $2^{-n}$
• Initial distributions become uniform asymptotically

This map models physical mixing processes and demonstrates how deterministic dynamics can homogenize distributions, providing a microscopic foundation for macroscopic diffusion and irreversibility.