10:06 AM
Simulations

1. Reaction-Diffusion

Visualizing the Gray-Scott equations: a mathematical model of complex Turing patterns where diffusion (DA, DB), feed rate (f), and kill rate (k) determine the structural evolution.

Governed by the Gray-Scott equations:

∂A∂t=DA∇2A−AB2+f(1−A)\frac{\partial A}{\partial t} = D_A \nabla^2 A - AB^2 + f(1 - A)
∂B∂t=DB∇2B+AB2−(k+f)B\frac{\partial B}{\partial t} = D_B \nabla^2 B + AB^2 - (k + f)B

2. The Lorenz Attractor

A 3D visualization of deterministic chaos. First studied by Edward Lorenz, these linked differential equations show how tiny changes in initial conditions result in drastically different, completely unpredictable outcomes—commonly known as the butterfly effect.

Governed by the Lorenz equations:

dxdt=σ(y−x)\frac{dx}{dt} = \sigma(y - x)
dydt=x(ρ−z)−y\frac{dy}{dt} = x(\rho - z) - y
dzdt=xy−βz\frac{dz}{dt} = xy - \beta z