I came across an interesting paper [1] about something called Labyrinth chaos.
The underlying form of the equations (in three dimensions, at least) which yields the chaos looks like:
dx/dt = f(x,y,z)
dy/dt = f(y,z,x)
dz/dt = f(z,x,y),
where the functions are all identical, except the variables are rotated. This, I believe, is called a circulant system [2]. The above-mentioned Labyrinth aspect occurs when f(x,y,z) is (I assume) periodic e.g. sine, cosine.
Götthans and Petržela give results, amongst others, for the case in which
f(x,y,z) = -ax - cos(by),
where a and b are constants.
Now, as it happens I now have an analog computer with trigonometric (amongst other) functions...
Interestingly, my understanding of the paper by Götthans and Petržela is that their approach is similar to mine: they have used a circuit implementation of the equations, with their integrations being dispatched using op amps with capacitor feedback, and their f(x,y,z) parts dealt with digitally.
(My 'special' functions are similarly dealt with digitally (urghh!!) by looking up MATLAB-calculated values from an EEPROM, sandwiched between an ADC and DAC - see my post of November 22, 2014...)
My first attempt at implementing Götthans and Petržela's equations directly,
dx/dt = - ax - cos(by),
dy/dt = -ay - cos(bz)
dz/dt = -az - cos(bx),
ended up with the integrators overloading (i.e. outputs greater than 10 V or less than -10 V). To overcome this problem I made the substitutions x = 2u, y = 2v, z = 2w, giving
du/dt = - au - cos(2bv)/2,
dv/dt = -av - cos(2bw)/2
dw/dt = -aw - cos(2bu)/2,
which fixed up the overloading issue. (The above factor of 2 was arrived at by guesswork. This whole area of scaling remains a bit of a black art...)
Here's my thoughts on paper as to how to connect the bits of the computer up:
...and here's how it was patched on the actual thing:
Integrator initial conditions were all set to zero volts.
Et voilà, results for three different values of constant b:
(In all cases the scope is 1 V/div in XY mode. I connected the scope via two times-2 amplifiers on the computer (not shown in above patch diagram), so x = 2u is connected to horizontal axis and y = 2v is connected to vertical axis.)
These look pretty much like the results in [1]: their figure 6 (negative cosine function) - although theirs are projections of the rotated 3D space, and there may well be some issues with signs (in my part - my b = 1 case seems to be in a different quadrant, compared to [1]).
References
[1] Experimental Study of the Sampled Labyrinth Chaos. Tomáš Götthans, Jiří Petržela, Radioengineering, Vol. 20, No. 4, December 2011.
[2] Elegant Chaos - Algebraically Simple Chaotic Flows. Sprott, Julien. C. World Scientific, 2010, p. 101.
The underlying form of the equations (in three dimensions, at least) which yields the chaos looks like:
dx/dt = f(x,y,z)
dy/dt = f(y,z,x)
dz/dt = f(z,x,y),
where the functions are all identical, except the variables are rotated. This, I believe, is called a circulant system [2]. The above-mentioned Labyrinth aspect occurs when f(x,y,z) is (I assume) periodic e.g. sine, cosine.
Götthans and Petržela give results, amongst others, for the case in which
f(x,y,z) = -ax - cos(by),
where a and b are constants.
Now, as it happens I now have an analog computer with trigonometric (amongst other) functions...
Interestingly, my understanding of the paper by Götthans and Petržela is that their approach is similar to mine: they have used a circuit implementation of the equations, with their integrations being dispatched using op amps with capacitor feedback, and their f(x,y,z) parts dealt with digitally.
(My 'special' functions are similarly dealt with digitally (urghh!!) by looking up MATLAB-calculated values from an EEPROM, sandwiched between an ADC and DAC - see my post of November 22, 2014...)
My first attempt at implementing Götthans and Petržela's equations directly,
dx/dt = - ax - cos(by),
dy/dt = -ay - cos(bz)
ended up with the integrators overloading (i.e. outputs greater than 10 V or less than -10 V). To overcome this problem I made the substitutions x = 2u, y = 2v, z = 2w, giving
du/dt = - au - cos(2bv)/2,
dv/dt = -av - cos(2bw)/2
dw/dt = -aw - cos(2bu)/2,
which fixed up the overloading issue. (The above factor of 2 was arrived at by guesswork. This whole area of scaling remains a bit of a black art...)
Here's my thoughts on paper as to how to connect the bits of the computer up:
...and here's how it was patched on the actual thing:
Integrator initial conditions were all set to zero volts.
Et voilà, results for three different values of constant b:
(In all cases the scope is 1 V/div in XY mode. I connected the scope via two times-2 amplifiers on the computer (not shown in above patch diagram), so x = 2u is connected to horizontal axis and y = 2v is connected to vertical axis.)
 |
| b = 1, a = 0.4 |
 |
| b = 5, a = 0.4 |
 |
| b = 2, a = 0.4 |
I particularly like the b = 5 case with the computer in FAST mode - the result on the scope looks like wisps of smoke; in normal mode (1 µF / 1 MΩ), the spot darting around on the scope looks very life-like - a fly or something...
And, finally...on paper it looks like (b = 5 case):
 |
| x is horizontal, y vertical, b = 5. |
References
[1] Experimental Study of the Sampled Labyrinth Chaos. Tomáš Götthans, Jiří Petržela, Radioengineering, Vol. 20, No. 4, December 2011.
[2] Elegant Chaos - Algebraically Simple Chaotic Flows. Sprott, Julien. C. World Scientific, 2010, p. 101.
No comments:
Post a Comment