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 = −ax − 4y − 4z − y²dy/dt = −ay − 4z − 4x − z²dz/dt = −az − 4x − 4y − x²- System
- Halvorsen
- a
- 1.4
- Start point
- -8.73, -3.56, 1.404
- Time step
- 0.004
- Integration steps
- 260,000
- Method
- RK4
- Lyapunov exponent λ
- 0.694
- View angle
- 85.3° / -4.1° / -14.3°
- Coverage (%)
- 35.5
λ = 0.694 > 0 means chaos: two paths starting 0.00000001 apart separate by a factor of e^(0.694·t). Their distance doubles every 1 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.
