Lorenz Attractor

Overview

The Lorenz attractor is a set of chaotic solutions to the Lorenz system, a system of three ordinary differential equations:

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

The system exhibits chaotic behaviour for certain parameter values. The classic parameters are $\sigma$ = 10, $\beta$ = 8/3, and $\rho$ = 28.

The trajectory never repeats itself and forms a butterfly-shaped attractor in 3D space.

This visualisation uses a 4th-order Runge-Kutta method to numerically integrate the system and render the resulting trajectory in 3D space. The colour of the trajectory varies based on time-step, creating a gradient effect.

Edward Lorenz developed the Lorenz system as a simplified mathematical model to understand atmospheric convection. The model describes how the three factors, the intensity of the convection (x), the temperature difference between the rising and falling air currents (y), and the distortion of the vertical temperature profile from a linear (z) changes over time.

The model was developed with the assistance of Ellen Fetter and Margaret Hamilton. Ellen Fetter performed the numerical simulations and created figures, while Margaret Hamilton helped with initial computations. The behavior of the system was found to be governed by the following equations:

The constants σ, ρ, and β are parameters representing physical properties of the system. σ here is known as Prandtl number, ρ is the Rayleigh number, and β relates to the physical dimensions of the fluid layer.

From a technical standpoint, the Lorenz system is nonlinear, aperiodic, three-dimensional, and deterministic. The Lorenz equations have been found to model behavior in a wide variety of systems, including lasers, dynamos, electric circuits, and even some chemical reactions.

Source: Wikipedia

Here's a cool video of the Lorenz attractor in action