Hyperchaos

Section 6.7.3 of my increasingly well-thumbed copy of Sprott [1] introduces coupled chaotic systems. For example we have two Lorenz systems:

Paper patching - this needs all four multipliers and
 six of the eight integrators.

In the above, k is the coupling between the two systems (k = 0 means no coupling - and the systems are independent).

And here's the yx-plot of the attractor (initial conditions as per Fig 6.27, [1]):

k = 0.6, R = 1, y vertical, x horizontal

Compares nicely with Sprott's figure 6.27 [1]. Given it's a coupled system, it's interesting to look at the projection of the attractor with (say) x (from 1st system) horizontal and v (from 2nd system) vertical:

xv projection with k = 0.6

As k gets bigger, there is evidently a tendency for the two systems to become synchronised [1] - for example with k = 0.68 we have the rather attractive (aesthetically speaking) attractor:

xv projection with k = 0.68

With k = 0.7 we have the limit cycle attractor:

xv projection with k = 0.7

Reference

[1] Elegant Chaos: Algebraically Simple Chaotic Flows, Julien Clinton Sprott, World Scientific, 2010, p. 157.