MATHEMATICS
The maths behind the picture
Every piece is computed, not drawn. These are the real equations and numbers from the program that made this exact piece.
dx/dt = σ(y − x)dy/dt = x(ρ − z) − ydz/dt = xy − βz- System
- Lorenz
- σ
- 10
- ρ
- 28
- β
- 2.666667
- Start point
- -6.353, -4.015, 27.645
- Time step
- 0.004
- Integration steps
- 260,000
- Method
- RK4
- Lyapunov exponent λ
- 0.907
- View angle
- 85.1° / -13.5° / 0.2°
- Coverage (%)
- 43.9
λ = 0.907 > 0 means chaos: two paths starting 0.00000001 apart separate by a factor of e^(0.907·t). Their distance doubles every 0.76 time units, so very soon they have nothing in common.
HISTORY
Where this idea comes from
Attractors
- 1890
Henri Poincaré, working on the three-body problem, first glimpses chaos: a tiny change at the start changes the whole outcome.
- 1963
Edward Lorenz at MIT reduces a model of the atmosphere to three equations and finds that weather cannot be predicted far ahead.
- 1971
David Ruelle and Floris Takens coin the term "strange attractor".
- 1972
Lorenz's talk "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?" gives the butterfly effect its name.
