Fig 1: *f*(*z*)=*cz*log(*z*) : (1) chosen again so that it has a crticial point at e^{-1 }to develop a parameter plane. The parameter plane with (2) a sub-region close to (e) showing quadratic dark heart Mandelbrot 'islands' and cuts in the complex plane caused by the original cut in the log function (3) a period 3 Julia set kerel from a period 3 bulb on (1). (4) A Julia set kernel from the cut black region (c) in (2) shows the same cuts in the Julia core (5). (6) Julia set kernel of the period (3) bulb in the dark heart (b) with its core enlarged in (7). (8) and (9) biomorphic Julia kernels associated with the cusps at (d) and (e).

Fig 2: *f*(*z*)=exp(-*z*^{2})+*c* : (1) Parameter plane with (2) inset region (c) displaying exotic dynamics. (3) and (4) Julia set kernels of the period 3 bulbs left and right of the period 3 bulb (a) and (b). (5) inset of (4). (6) A dark heart Mandelbrot 'island' off one of the main bulbs in 1 showing an intersection region of exotic dynamics (left). (70 and enlarged field of the same region showinf several dark hearts.

Fig 3 *f*(*z*)=*czz ^{z }*=

**Quicktime movie of f(z)=cz^{(z+k)}, k=0, ..., 1**

Fig 4 *f*(*z*)=Γ(*z*) : (Left) Colour plot of the function by its real and imaginary parts. (Centre) Parameter plane of *f*(*z*)=Γ(*z*)+*c* (compare the Riemann-Zeta function. (Right) Julia set of *f*(*z*)=Γ(*z*) showing internal basins of attraction to *z*=1.

Fig 5 below extends this comparison and the complex interaction of successive critical points further, by adding the levels of all 8 critical points outside
, so that the interaction of each of the critical points can be seen without the masking effect of the method of fig 34. This demonstrates that, in addition to the sensitive regions highlighted in fig 34 there are other regions in which the bifurcations of several critical points come together to create overlapping bifurcations. The Julia structure corresponding to one particular intersection of the bifurcation bulbs of three critical points is examined to show the variety of bifurcating quadratic kernels.

Fig 5: The function
illustrated by adding the levels of eight critical points outside
shows the interaction of parameter planes of several critical points in three focal regions (a, b, c). Region (b) is then expanded and the Julia structure corresponding to the intersection of the bifurcation curves (b2) is shown right with expanded views of sectors (1, 2, 3, 5, 6, 8, and 10) showing the quadratic kernels corresponding to the interaction locus in (b2).

**Chaotic Processes and Discrete Iteration**

All computer methods suffer from numerical over/underflow and the incapacity of any simulation to accurately approximate a dynamical process, which is sensitive to its initial conditions. Thus Mandelbrot originally thought 'his' set was disconnected into islands until Douady and Hubbard's, conformal proof of its connectedness demonstrated the contrary. However these problems can lead to intriguing issues of computational complexity.

The function
shows how sensitive computer processes can be to underflow and overflow, resulting in discrete artifacts similar to a cellular automaton, which can be even more beautiful and complex than the underlying process. The underlying degree of the function is 1, since the first order degree of the cosine 2 is cancelled by the degree 1/2 of the root in composition. The process thus becomes highly sensitive to floating point over/underflow at bifurcation points, particularly the principal explosion point.

Fig 6: Cosine root principal explosion in 16384 iterations to show the initial stages of the explosion, using the arctan formula. As can be seen, in contrast to figure 32, the initial explosion is characterized as an atomic cosmological 'big-bang' filling parabolic space with extremely high iteration number structures at the limits of complexity resolution, followed by successively slower explosion waves of successively simpler level set structure, as the process progresses. These suggest associations both with quantum chaos and with cellular automata, some of which, on the 'edge of chaos' in discrete system terms, can act as universal computers.

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Quicktime movie of the above explosion

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In taking
, we can proceed in several ways. In fig 41 we have used
the apparently simplest route. However this involves both calculating a double square root
and using the transcendental arctan to halve the angle
making allowance for singularities.

Alternatively, we can proceed directly:

At this point we would be tempted to use
, but this is liable to over/underflow error and singularity, resulting in gross divergence at the explosion point. Instead we can define:

This method gives the series of images in fig 33 which coincide with those for the other functions.

A hint of why the phenomena of fig 6 may be happening can be seen from fig 7, where discrete effects emerge from the underflow of computations of a radial wave function under recursive dilation of the origin.

Fig 7: Super-attracting basin of
becomes a Moire pattern when a radial wave function about the origin is used, as the inverse process is recursively dilating the origin.