Endo - iterate an endomorphism of the plane
----

Written by Ronald Joe Record ([email protected]) 17 Jan 1992

INTRO
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The basic idea is to iterate an endomorphism of the plane (R x R).
Each point (x,y) is mapped to (f(x,y),g(x,y)). The new point is
then mapped to (f(f(x,y),g(x,y)),g(f(x,y),g(x,y))) and so on. Each
point is then colored according to how many iterates it took to get
within some "delta" of a previous iterate (e.g. a periodic orbit is
detected). Lyapunov exponents can be calculated and displayed for a 
region of parameter space. Phase portraits can be constructed and
histographic data displayed. Finally, critical curves and their iterates
may be displayed (curves for which the determinant of the Jacobian is zero).

ACKNOWLEDGEMENTS
----------------

The Winter 1992 seminar at UCSC led by Dr. Ralph Abraham was the
inspiration for the study of iterated endomorphisms of the plane.

Techniques for calculating the critical arcs and absorbing regions
were drawn from Laura Gardini of the Istituto di Scienze economiche,
Universita' di Urbino. Studies of various maps of the plane were used
including papers by Jon von Neumann, Graciela Chichilnisky, John 
Guckenheimer, G. Oster, A. Ipaktchi, Wulf Gaertner, Jochen Jungeilges,
Ralph Abraham, B. P. Wood, A. J. Lichtenberg, Z. You, E.J. Kostelich,
J. A. Yorke, R. Brown, L. Chua, K. Judd, A. I. Mees, K. Aihara, and
M. Toyoda.

BUILD
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To build the endo binary, either use the Imakefile or one of the sample 
makefiles - Makefile.ODT or Makefile.OSF. Makefile.ODT is a sample makefile
used to build endo on an SCO ODT system. Makefile.OSF was used as a makefile
on a DECstation 3100 running OSF/1. The manual page can be formatted by typing 
"nroff -man endo.man > endo.doc".

INSTALL
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To install endo, copy the endo binary to the desired location (the sample
makefiles put it in /usr/local/bin/X11).
Copy the formatted man page to wherever you keep your local doc (i use
/usr/man/man.LOCAL for imaging software), then add that location
to your MANPATH.

Some "interesting" runs of endo are included as simple shell scripts in the
"params" subdirectory.


CUSTOMIZING
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To add a map to the array of selectable endomorphisms, follow the following
steps :

	1. Edit endo.h and add the pair, double, PFP, PFD, and Mapnames
	   declarations. Follow the example set by the "standard" map.
	2. Edit maps.c and add the map and derivative function definitions.
	   Again, you can follow the example set by standard() and dstandard().
	3. Edit params.h adding the numerical values to use in the 
	   amins, aranges, bmins, branges, pmins, pmaxs, and defparms arrays.
	   In each case, when adding the nth map, you will be adding the nth
	   value in the array (which may itself be an array).
	4. Edit defines.h, incrementing NUMMAPS and increasing NUMDEFS by 2.
	5. Edit info.c, adding a string representation of the function to
	   Mapdefs[] and an entry in the numparams array indicating how many
	   parameters the map has.
	6. make clean; make tags; make

Ideas, comments, additions, deletions, suggestions, bug reports, code review,...
e-mail Ronald Record at [email protected]