At 21:24 18/05/2002, Andrew Coppin wrote:
I also did some work with POVRay, drawing graphs for forwards and backwards iterations of the quadratic mapping (with Z-depth indicating iteration count), trying to work out where those complex patters derrive from. I mean, z => z^2 is a very simple function; it's just a big troff! (Oh dear... too much UNIX :S) The Julia set reflects this - it's a circle. Adding the c term only moves the center of the thing... and yet this creates amazingly complicated patterns. Why?
I could suggest further experiments with Fractint; specifically, the mandelcloud type. Pick a high interval rate (something like 200 or so) and a low iteration rate - actually, start with 2 and work up from there. (And I wonder; what about fractional iterations? What'd be a sensible definition of those? See what's going on in between iterations...) This is something I'd like to do with any user-defined formula. I have several that are made for this sort of treatment that (I think) lose much of their interest when rendered as an escape-time fractal, since the interest is in the dynamics not the outcome; the bailout in such situations tends to be an ugly hack with little justification. Some of my Chebyshev series have this property. But what you're describing is something that often treated in mathematics under the heading of bifurcation theory. A very approachable text (for those whom the author describes as requiring "a working knowledge of basic undergraduate mathematics" is John L. Casti's twin-volume _Reality Rules_. Chapter 2, for instance, has the sections: 1. The Classification Problem 2. Smooth Functions and Critical Points 3. Structural Stability and Genericity 4. Morse's Lemma and Theorem 5. The Splitting Lemma and Theorem 6. Determinacy and Codimension 7. Unfoldings 8. The Thom Classification Theorem 9. Electric Power Generation 10. Bifurcations and Catastrophes 11. Harvesting Processes 12. Estimation of Catastrophe Manifolds 13. Forest Insect Pest Control 14. The Catastrophe Controversy 15. Mappings 16. Dynamical Systems, Flows and Attractors 17. Bifurcation of Vector Fields 18. Stochastic Stability and the Classification of Vector Fields I'll hold off on the full contents of both volumes unless they're asked for :) I'm not going to say that all is finished in the subject. Knickers with avengence. I'm just offering up these suggestions for tools and lines of attack and expect to see lots of pretty pictures as a result. Think of me more as an arms dealer than a peacekeeping force :) Morgan L. Owens "Scared, yet?"