Strange Attractors

Programming as a creative endeavour

Strange attractors are mathematical sets that emerge from chaotic systems, revealing intricate patterns resulting from the sensitive dependence on initial conditions.

At its core, a strange attractor is a set of numerical values towards which a system evolves over time which can be used to render a unique and often intricate pattern. These attractors emerge from the mathematical equations that govern chaotic systems, where small variations in initial conditions lead to vastly different outcomes. The beauty of strange attractors lies in their ability to reveal a hidden order within apparent randomness.

Several mathematical equations give rise to popular strange attractors, each producing distinct and visually striking patterns. Here are a few notable examples along with their corresponding equations:

Lorenz Attractor:

x˙=σ(yx)y˙=x(ρz)yz˙=xyβz\begin{align*} \dot{x} & = \sigma(y - x) \\ \dot{y} & = x(\rho - z) - y \\ \dot{z} & = xy - \beta z \end{align*}

Parameters: σ\sigma, ρ\rho, and β\beta control the behavior of the system.

The Lorenz attractor, discovered by Edward Lorenz, exhibits a butterfly-like shape and is a classic representation of deterministic chaos. Adjusting the parameters results in a variety of visually captivating outcomes.

Here is a simple demo

Rossler Attractor:

x˙=yzy˙=x+ayz˙=b+xzcz\begin{align*} \dot{x} & = -y - z \\ \dot{y} & = x + ay \\ \dot{z} & = b + xz - cz \end{align*}

Parameters: aa, bb, and cc determine the system's behavior.

The Rossler attractor produces a set of looping curves, creating a visually appealing and intricate pattern reminiscent of chaotic motion.

Here is an interactive demo

Hénon Map:

xn+1=1axn2+ynyn+1=bxn\begin{align*} x_{n+1} & = 1 - ax_n^2 + y_n \\ y_{n+1} & = bx_n \end{align*}

Parameters: aa and bb influence the map's characteristics.

The Hénon map is a discrete-time dynamical system that generates visually compelling patterns resembling a scattered collection of points.

By assigning variables from the equations to the coordinates of a graphical representation, such as a pixel on a screen, intricate patterns emerge. The iterative nature of the equations allows for the creation of evolving, dynamic visuals, making each piece of art unique.