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.