Tetration Chaos

Here I've taken one of Sprott's simple three-dimensional chaotic flows with a quadratic nonlinearity (the system SQF, from table 3.1 reference [1]),

and replaced the quadratic nonlinearity with the tetration operator x^x  to give the system:

.

The 2nd order tetration is realised on the analogue computer by multiplying the logarithm of |x| with |x| and then taking the exponential, this is all patched up as shown below:

   

Initial conditions were (0,-0.5,0). The following pictures show the results from the analogue computer: pen plots on the left (600 seconds' worth), oscilloscope traces on the right (with computer in FAST mode):

XZ-plane (left is pen plot (0.5 V/cm), right from oscilloscope (1 V/div))

XY-plane

YZ-plane (plotter)

YZ-plane (oscilloscope)

And here are some MATHCAD results:

MATHCAD results (∆t = 0.00025) 

In passing, it's noted that the system has two equilibrium points, namely (-0.3463233..., -0.6926467..., 0.6926467...) and (-2, -4, 4), and the divergence of the system is equal to -1/2 and the system is dissipative.


It is apparent (and presumably not surprising) that many of the other chaotic systems remain chaotic when (for example) quadratic terms are replaced with |x|^|x|, e.g. Nosé–Hoover.

Reference

[1] J.C. Sprott, Elegant Chaos - Algebraically Simple Chaotic Flows, World Scientific, 2010, p. 70.