# Essays/Lorenz Attractor

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OpenGL Demo script File:Lorenz 3d.ijs

The Lorenz attractor was obtained by Edward Lorenz in 1963 when simplifying a system of fluid convection flow equations, and since then captured attention of both serious and popular science. The recognizable image is the trajectory of a solution of a dynamical system with the following variables and parameters.

X convective intensity Prandtl number Y vertical/horizontal temperature difference Rayleigh number Z vertical temperature deviation geometric factor

In J this can be expressed as follows File:Lorenz j504.ijs

'`X Y Z'=: (0&{)`(1&{)`(2&{) ' s r b'=: 10 , 28 , 8%3 dx=: s*(Y - X) dy=: (X * r - Z) - Y dz=: (X*Y) - b*Z dt=: 0.005 I=: + dt * dx,dy,dz

## Quadratic Map

The system can also be expressed as a three-dimensional quadratic map.

N=: ([: ; i. <@}."0 1 i.@,~) 4 NB. upper triangle indices NB. xx xy xz x1 yy yz y1 zz z1 11 NB. ;,&' '&.> N { ,<@,"0/~'xyz1' A=: ,:0 0 0 ,(-s),0 0 ,s, 0 0 0 A=: A,0 0 _1 ,r, 0 0 _1 0 0 0 A=: A,0 1 0 0 0 0 0 0 ,(-b),0 dL=: A +/ . * N { [: , [: */~ ,&1 J=: + dt * dL

The image was obtained with the Stereo Plot utility.

## See Also

- Essays/Rössler Attractor by MasatoShimura
- WikiPedia:Lorenz_attractor, Wikipedia
- MathWorld:LorenzAttractor, also MathWorld:QuadraticMap, Mathworld
- Paul Bourke -- view and listen!

Contributed by Oleg Kobchenko.