• Equations

The sea model uses the wave equation:

d²ψ/dt² = c² ∇² ψ

The strange attractors are drawn using the following equations:

The Lorenz equations:

dX/dt = σ ( Y – X )

dY/dt = X ( ρ – Z )

dZ/dt = X Y – ß Z

The Rössler attractor:

dx/dt = – y – z

dy/dt = x + ay

dz/dt = b + z ( x – c )

The Mandelbrot set is drawn using the map:

Z_n+1 = Z_n² + Z₀

The photos of smoke could be modelled approximately using the Navier-Stokes equations, which result from Newton's laws being applied to a continuous fluid:

ρ ( ∂u/∂t + u⋅∇u ) = – ∇p + ∇⋅{ μ (∇u + (∇u)^T – ⅔ ( ∇⋅u ) I } + ρ g

Flames rolling slowly in a wood burner & smoke rising from a candle, following the Navier-Stokes equations, but also affected by temperature changes during methane burning or heating are very beautiful:

The equations for the strange attractors and the Navier-Stokes equations can be written down but not solved exactly.

A computer program can be used to solve many equations by approximating values such as

dY / dX

which occur in the above equations.

dY / dX indicates how much a value Y changes when another value X changes.

The simplest computer code for calculating this would be of the form

( Y at (X + 1) - Y at X ) ÷ 1

Although the Navier-Stokes equations are 'simply' Newton's laws applied to a 3 dimensional fluid they are very difficult to solve. Often computer approximations produce infinite values or zero. This is characteristic behaviour of any chaotic system and is commonly demonstrated by the butterfly effect.

A very good numerical Navier-Stokes model in 2 dimensions is at Oliver's simple fluid dynamics simulator