Polynomial Attractor (Sprott 3D System)

Overview

In 1993, physicist and chaos pioneer J.C. Sprott at the University of Wisconsin-Madison asked a bold question: How many distinct strange attractors exist in the general mathematical space of quadratic differential equations?

Using automated Monte Carlo computer search algorithms, Sprott searched millions of randomly selected coefficient combinations. His search unveiled hundreds of brand-new chaotic attractors — from simple ribbon scrolls to intricate multi-winged organisms — proving that strange attractors are not rare exceptions in mathematics, but rather a ubiquitous natural feature of non-linear systems.

The Mathematical Framework

The general 3D quadratic polynomial flow is represented compactly as:

$\frac{d\vec{v}}{dt} = \mathbf{C} \cdot \vec{T}(\vec{v})$

where $\vec{v} = (x, y, z)^T$, $\vec{T}(\vec{v})$ is the 10-element vector of all constant, linear, and quadratic combinations:

$\vec{T}(\vec{v}) = \begin{bmatrix} 1 & x & y & z & x^2 & xy & xz & y^2 & yz & z^2 \end{bmatrix}^T$

and $\mathbf{C}$ is a $3 \times 10$ matrix containing 30 real-valued coefficients.

Mining Chaos with Strange Attractor XYZ

Our visualizer includes an automated Lyapunov Miner that scans this 30-dimensional parameter space:

1. Initial Seed: Random coefficients are assigned within the range $[-1.2, +1.2]$.
2. Integration & Stability Testing: Runge-Kutta integration evaluates whether orbits diverge to infinity or collapse into a static point.
3. Lyapunov Exponent Filter: The algorithm calculates the largest Lyapunov exponent $\lambda_1$. If $\lambda_1 > 0$ and the trajectory remains bounded within a finite volume, a new strange attractor is born.
4. Artistic Rendering: Millions of points are integrated with sub-pixel jitter, volumetric light accumulation, and custom color gradients to produce high-resolution fine art prints.