I have come across another interesting paper [1] which has several examples of chaotic systems. I've taken to working my way through these systems, beginning with the classic Lorenz attractor.
Amongst other things, the paper gives a set of equations (2), which are derived by applying a linear transformation to Lorenz's original equations, giving
dx/dt = α(y - x)
dy/dt = -xz -γy
dz/dt = xy - β/α - βz ,
where the new dimensionless parameters are given by α = √(σ/r), γ = 1/√(σr), and β = b/√(σr).
Using Lorenz's (classic butterfly) σ = 10, r = 28, and b = 8/3 values, gives α = 0.598, β = 0.159, and γ = 0.0597. This approach is useful because it allows direct patching on the 10 V computer.
I patched the above system of equations, with the above parameter values, like this:
which looked like this on the actual computer:
The oscilloscope display shows the output with the computer is FAST mode; but it's more fun directing the output to an XY plotter:
...and here are the plots (left is xz-plane, right is xy-plane).
Looking at these plots - especially when on paper, I cannot help but wonder if this type of thing (i.e. analog computer rendering of chaos) had any influence on British 20th century art (i.e. modernism and in particular modern sculpture - e.g. Barbara Hepworth)?
[1] Sprott, J. C., 'Simplifications of the Lorenz Attractor', Nonlinear Dynamics, Psychology, and Life Sciences, 2009, Vol. 13, No. 3, pp. 217 - 278.
Amongst other things, the paper gives a set of equations (2), which are derived by applying a linear transformation to Lorenz's original equations, giving
dx/dt = α(y - x)
dy/dt = -xz -γy
dz/dt = xy - β/α - βz ,
where the new dimensionless parameters are given by α = √(σ/r), γ = 1/√(σr), and β = b/√(σr).
Using Lorenz's (classic butterfly) σ = 10, r = 28, and b = 8/3 values, gives α = 0.598, β = 0.159, and γ = 0.0597. This approach is useful because it allows direct patching on the 10 V computer.
I patched the above system of equations, with the above parameter values, like this:
which looked like this on the actual computer:
The oscilloscope display shows the output with the computer is FAST mode; but it's more fun directing the output to an XY plotter:
...and here are the plots (left is xz-plane, right is xy-plane).
Looking at these plots - especially when on paper, I cannot help but wonder if this type of thing (i.e. analog computer rendering of chaos) had any influence on British 20th century art (i.e. modernism and in particular modern sculpture - e.g. Barbara Hepworth)?
[1] Sprott, J. C., 'Simplifications of the Lorenz Attractor', Nonlinear Dynamics, Psychology, and Life Sciences, 2009, Vol. 13, No. 3, pp. 217 - 278.
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