Wednesday, July 23, 2014

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.






Friday, July 18, 2014

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.

Thursday, July 17, 2014

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...




Saturday, July 12, 2014

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.


Friday, July 4, 2014

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).