Modeling Chaos: Strange Attractors in 3D Phase Space

01/ What is a Strange Attractor?

In the mathematical study of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve over time, regardless of the system's starting parameters. While simple attractors typically collapse into stable points (steady-state equilibrium) or closed loops (periodic limit cycles), a Strange Attractor is fundamentally different.

Characterized by a non-integer, fractal dimension, strange attractors display extreme sensitivity to initial conditions—a phenomenon colloquially known as the "Butterfly Effect". Even a micro-variance in starting state (e.g., $10^{-16}$) will cascade exponentially, leading to entirely divergent trajectories while still remaining bound within the global attractor's geometry.

The term was coined in 1971 by physicists David Ruelle and Floris Takens in their groundbreaking paper on the nature of turbulence, proposing that chaotic attractors are the mathematical engine behind fluid dynamics.

Interactive Phase Space Preview

2D Projection of real-time particle integration (RK4)

dt: 0.005 | x: 0.00 y: 0.00

02/ High-Fidelity 3D Simulation

To explore these structures in three dimensions, we built attractors-rs, a desktop and WebAssembly simulation engine written in Rust. The engine integrates millions of particles in parallel using hardware-accelerated vertex buffers, allowing real-time parameter tweaking.

Initializing attractors-rs.wasm...

WASM Hardware Accelerated Interactive 3D Engine

Figure 1: The attractors-rs engine running live via WebAssembly. Click and drag to rotate the camera.

For numerical integration, standard Euler integration ($x_{t+1} = x_t + f(x_t) \Delta t$) is too unstable and prone to accumulated errors over time, causing paths to drift off the attractor completely. Therefore, the engine implements the Runge-Kutta 4th Order (RK4) method.

Numerical Integration Scheme: RK4

The Runge-Kutta 4th Order integrator approximates the next point by calculating weighted averages of four increments across the interval $\Delta t$:

$$k_1 = f(t_n, y_n)$$

$$k_2 = f\left(t_n + \frac{\Delta t}{2}, y_n + \Delta t \frac{k_1}{2}\right)$$

$$k_3 = f\left(t_n + \frac{\Delta t}{2}, y_n + \Delta t \frac{k_2}{2}\right)$$

$$k_4 = f(t_n + \Delta t, y_n + \Delta t k_3)$$

$$y_{n+1} = y_n + \frac{\Delta t}{6} (k_1 + 2k_2 + 2k_3 + k_4)$$

This fourth-order approximation achieves an error bounds of $\mathcal{O}(\Delta t^4)$, maintaining trajectory coherence for chaotic structures over thousands of frames.

03/ Equations of the 11 Strange Attractors

Below are the ordinary differential equations (ODEs) and canonical parameters of the 11 chaotic systems.

1. Lorenz Attractor

LORENZ (1963)

$$ \begin{aligned} \frac{dx}{dt} &= \sigma (y - x) \\ \frac{dy}{dt} &= x (\rho - z) - y \\ \frac{dz}{dt} &= x y - \beta z \end{aligned} $$

Standard Parameters σ = 10.0, ρ = 28.0, β = 8/3 (2.667)

2. Thomas' Attractor

THOMAS (1999)

$$ \begin{aligned} \frac{dx}{dt} &= \sin(y) - b x \\ \frac{dy}{dt} &= \sin(z) - b y \\ \frac{dz}{dt} &= \sin(x) - b z \end{aligned} $$

Standard Parameters b = 0.208186

3. Chen Attractor

CHEN (1999)

$$ \begin{aligned} \frac{dx}{dt} &= a (y - x) \\ \frac{dy}{dt} &= (c - a) x - x z + c y \\ \frac{dz}{dt} &= x y - b z \end{aligned} $$

Standard Parameters a = 35.0, b = 3.0, c = 28.0

4. Dadras Attractor

DADRAS (2009)

$$ \begin{aligned} \frac{dx}{dt} &= y - a x + b y z \\ \frac{dy}{dt} &= c y - x z + z \\ \frac{dz}{dt} &= d x y - e z \end{aligned} $$

Standard Parameters a = 3.0, b = 2.7, c = 1.7, d = 2.0, e = 9.0

5. Rössler Attractor

RÖSSLER (1976)

$$ \begin{aligned} \frac{dx}{dt} &= -y - z \\ \frac{dy}{dt} &= x + a y \\ \frac{dz}{dt} &= b + z (x - c) \end{aligned} $$

Standard Parameters a = 0.2, b = 0.2, c = 5.7

6. Halvorsen Attractor

HALVORSEN

$$ \begin{aligned} \frac{dx}{dt} &= -a x - 4y - 4z - y^2 \\ \frac{dy}{dt} &= -a y - 4z - 4x - z^2 \\ \frac{dz}{dt} &= -a z - 4x - 4y - x^2 \end{aligned} $$

Standard Parameters a = 1.89 (also 1.4)

7. Rabinovich-Fabrikant

RABINOVICH (1979)

$$ \begin{aligned} \frac{dx}{dt} &= y (z - 1 + x^2) + \gamma x \\ \frac{dy}{dt} &= x (3z + 1 - x^2) + \gamma y \\ \frac{dz}{dt} &= -2z (\alpha + x y) \end{aligned} $$

Standard Parameters α = 0.14, γ = 0.10

8. Three-Scroll Unified

PAN & ZHOU (2010)

$$ \begin{aligned} \frac{dx}{dt} &= a (y - x) + c x z \\ \frac{dy}{dt} &= b x - x z + f y \\ \frac{dz}{dt} &= d z + x y - e x^2 \end{aligned} $$

Standard Parameters a=32.48, b=45.84, c=1.18, d=0.13, e=0.57, f=14.7

9. Sprott (Sprott LinZ)

SPROTT (2014)

$$ \begin{aligned} \frac{dx}{dt} &= y + a x y + x z \\ \frac{dy}{dt} &= 1 - b x^2 + y z \\ \frac{dz}{dt} &= x - x^2 - y^2 \end{aligned} $$

Standard Parameters a = 2.07, b = 1.79

10. Four-Wing Attractor

WANG & SUN (2009)

$$ \begin{aligned} \frac{dx}{dt} &= a x + y z \\ \frac{dy}{dt} &= b x + c y - x z \\ \frac{dz}{dt} &= -z - x y \end{aligned} $$

Standard Parameters a = 0.2, b = 0.01, c = -0.4

11. Langford (Aizawa) Attractor

AIZAWA (1982)

$$ \begin{aligned} \frac{dx}{dt} &= (z - b)x - d y \\ \frac{dy}{dt} &= d x + (z - b)y \\ \frac{dz}{dt} &= c + a z - \frac{z^3}{3} - (x^2 + y^2)(1 + e z) + f z x^3 \end{aligned} $$

Standard Parameters a = 0.95, b = 0.7, c = 0.6, d = 3.5, e = 0.25, f = 0.1