Smorgasbord #1 Rikitake Dynamo

The Rikitake system, which apparently serves as a model for the reversal of polarity of the Earth's electromagnetic field, is written:

Interestingly, I believe this predates the Lorenz attractor.

I wired up the computer to model the above; phase portraits of the Rikitake attractor for μ = 2, α = 5, x₀ = 0, y₀ = 0.1 and z₀ = 0 are shown below (1 V/cm, scope in XY-mode), together with results from reference [1]. I've tried to adjust my results to match the scales used in [1]. Not sure why my results show a preponderance for the right hand side - I think it may be because the analog computer solution decays (over time) due to the integrators not being completely error free: by the time the solution is ready to move across to spiral around on the left hand side, errors have possibly built up, destroying the solution.

[1] Ihsan Pehlivan and Yulmaz Uyaroglu, 2007. Rikitake Attractor and its Synchronization Application for Secure Communication Systems. Journal of Applied Sciences, 7: 232-236.

Bessel Attractors

I was looking at multi-scroll attractors - based on the 3-D linear autonomous system mentioned in [1, equation (11)],

More about the multi-scroll attractors in a later post.

In the meantime I wondered - as you do - what would happen if the function on the right hand side of dz/dt in the above were a Bessel function (first kind - and indeed the only kind on my computer!)...as an aside I am fascinated by an age in which each differential equation effectively had its own named function (what we would now just call a solution)...years ago I managed to get incomplete Anger functions into my thesis...

So, in jerk form we have:

A trivial re-arrangement gives

which has the form of Sprott's 'memory oscillator'...[2, page 82]:

although in my case the first derivative has a non-unity coefficient, b. After integrating the above, term by term, with respect to time, I think it would be reasonable to associate b with some measure of the square of the undamped angular frequency of the oscillator. (It would be interesting to vary b...).

Below are my analogue computer results obtained for a = 0.4, b = c = 0.8, q = 8, and n = 0, 1, and m = 1, 2 and 5:

(Here the top row is at 5 V/div; bottom row is 1 V/div; all others 2 V/div).

The J₁(5x) are worthy of being plotted on actual paper:

XY-plane

XZ-plane

YZ-plane

The last one looks almost composed of cylindrical orbits about a straight axis. Bessel functions are cylinder functions after all...

References

[1] Jinhu Lu, Guanrong Chen, Xinghuo Yu, H. Leung, ' Generating multi-scroll chaotic attractors via switching control', 5th Asian Control Conference, 20-23 July 2004 pp. 1753 - 1761.

[2] Julien C. Sprott , Elegant Chaos: Algebraically Simple Chaotic Flows, World Scientific Publishing Company, 2010.

Beyond Rössler

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.)

b = 1, a = 0.4

b = 5, a = 0.4

b = 2, a = 0.4

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]).

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.

Front and Back

Coefficient Potentiometers Rack



Special Functions Rack



Integrator Rack



Control Rack

Wire

Some photos of the special functions rack, which is now - after several mammoth wiring sessions - complete.

 

Knot Theory

Cannot resist messing about - now special functions rack nears completion - here I have put sin(x) on the horizontal axis of the scope, and Bessel function J₁(x) on vertical axis (here x(t) is a triangle wave)...yields a Lissajous-like figure...looks something like a figure eight knot...

5 V per division (XY-mode)

Front Panel Furniture

Orange banana jacks arrived from US and wiring of the front panel furniture for the special functions rack continues...with a bit of testing as I go along...