Götthans-Petržela Oscillator

Previously (Beyond Rössler, 1st February 2015), I looked at:

dx/dt = - ax - cos(by),

dy/dt = -ay - cos(bz)

dz/dt = -az - cos(bx).

An interesting variant [1] is to take the sign of the trigonometric function - in effect replacing the (co)sinusoidal right hand side terms with square waves:

dx/dt = - ax - sign[cos(by)],

dy/dt = -ay - sign[cos(bz)]

dz/dt = -az - sign[cos(bx)].

So as to compare results with those in reference [1], I went for the sine function:

dx/dt = - ax + sign[sin(by)],

dy/dt = -ay + sign[sin(bz)]

dz/dt = -az + sign[sin(bx)].

Here's my patching for the above:

(My computer only has two x > y functions - these are used for two of the signums. The remaining one is cooked up using the output x < 0 of the spare special function and an electronic switch. In hindsight, perhaps a minimum of three of each functionality would be nearer the mark - given that we live in a three-dimensional world.)

a = 1, b = 10. y axis is vertical, x horizontal, 2 V /division.

The above compares with (the xy-plane) of Figure 4 of reference [1]. Initial conditions are x = 0.1, y = 0, z = 0.

Setting a = 2 gives the following:

(I've fiddled with the colours in the above to agree with [1]: I am guessing that, in Figure 8 of reference [1], the red plot is the xy-projection, the blue is the xz projection.)

Finally, setting a = 0.1 gives an obvious clipping problem - but in the region for which |x| and |y| < 10 V, I get the same sort of structure as in Figure 9 of [1]:

y vertical, x horizontal, 5 V / div.

Interesting stuff. And probably the most complicated problem I've patched up:

Postscript 

A benefit of the analog computer approach is that it's entirely trivial to try out ideas like this:

where n = 1, 2, ...,6 (at least that's what's available on my computer!).

I (naively) thought that the above might put a cat amongst the pigeons, so to speak - i.e. yield a more 'complicated' attractor - because - at least for odd values of n - we have things like:

...but in an annoying face palm moment it's obvious that:

durhhhh - and so I got the same attractors as before.

(I had a look at these attractors sans the sign functions - but they looked pretty much like the ones for n = 1. I guess the coefficients in front of the 'higher frequency' terms in the above expansions weaken the effect / contribution of those bits.)

Reference

[1] Petrzela, J., Gotthans, T., Hrubos, Z. Analog implementation of Gotthans-Petrzela oscillator with virtual equilibria, Radioelektronika (RADIOELEKTRONIKA), 21st International Conference, 19-20 April 2011, pp. 1-4.