Cubism

Scaling a Simple Jerk System

In this context jerk is the rate of change of acceleration...

Presumably this is all well known to most people, but I find it interesting that (certain?) chaotic systems can be written in jerk form...

dz/dt = J(x,y,z)

where J is (some?) function of x, dx/dt, and d² x/dt² i.e. with y = dx/dt and z = dy/dt = d² x/dt² .

For example (Sprott [1], equation (10)):

d³x/dt³ = -a d² x/dt² + (dx/dt)² -x .

Which - as Sprott says - is chaotic for 'a small range of the single parameter a around 2.02'.

Writing out the three equations gives something to wire up:

dx/dt = y

dy/dt = z

dz/dt = -a z + y² - x .

I tried patching these up directly - but the thing overloaded - so nothing for it but to have a go at scaling. After looking in several books (and being not much the wiser - I generally need to construct things for myself using more, smaller, steps than most books seem to offer), I decided to just write down:

u = x/2, v = y/2 and w = z/2 → x = 2u, y = 2v and z = 2w .

Hence

d(2u)/dt = 2v → du/dt = v;

d(2v)/dt = 2w → dv/dt = w;

d(2w)/dt = -2wa + (2v)² → dw/dt = -aw + 2v² - u.

I guess it's obvious (in hindsight) that it's the non-linear bit of the equations which ends up with the scaling. I patched these 'scaled' equations up:

One nice things is the elegance of the patching, in the sense that the integrators form a simple chain.

In FAST mode the computer produced this the following. This proved stable - but only for a small range of values for parameter a:

Slowing things down (i.e. 1 μF integrating capacitors / 1 M ohm input resistors) gave the following pen plot - but note it goes awry after half a dozen 'orbits'...

The final part of [1] alludes to jerky Lorenz-like systems, which involve - amongst other things - functions like tan and hyperbolic sine :) But these will have to wait awhile, whilst the final(?) rack of my computer gets some attention...this - it is planned - will furnish sine, cosine, tangent, and their hyperbolic cousins...

[1] Sprott, J. C., 'Simplifications of the Lorenz Attractor', Nonlinear Dynamics, Psychology, and Life Sciences, 2009, Vol. 13, No. 3, pp. 217 - 278.

Another Day, Another Attractor

...a frisky* little quadratic nonlinearity attractor, found in [1]...

dx/dt = -z,

dy/dt = -x² - y,

dz/dt = 1.7 + 1.7x + y.

(Initial conditions 1, -0.8 and 0.)

* evidently very sensitive to the 1.7 multiplicative factor on the x term.

[1] Elegant Chaos: Algebraically Simple Chaotic Flows (Julien C. Sprott, World Scientific, 2010), Fig 3.9, page 71, SQM.

Chaotic Behavior and Shil’nikov Homoclinic Orbits

My exploration of chaos continues...equation (5) from Sprott's paper:

dx/dt = -y - z

dy/dt = x

dz/dt = a(y - y²) - bz , with a = b = 0.5.

This was interesting because the first attempt ended in failure (i.e. overload condition on all the amplifiers); a bit of sleuthing uncovered a typo in equation (5) for the dz/dt term (the final x should, in fact, be a z)...and voila...

It is the Rössler prototype-4 system, apparently. The picture below shows how I figured out the patching (I have taught myself to mark up each connection on the diagram with a highlighter pen, as I patch that lead physically on the computer...this avoids my getting into a complete mess...). Also, you can see the crossed out part of the first, wrong, attempt...

Self Propelled Flowers

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.

A minor bit of circuitry

Just for the record...the +/- 21 V supply is available on the external bus...

...and there's 12 V for a cabinet fan (maybe - if needed).