How To Draw Orbit Diagram In Mathematica

We'll begin with a note about Bifurcation and Orbit diagrams. Plots of these two diagrams for the family of functions f_c(x) = xⁱⁱ + c appear below. The shadowy curves sweeping up and downwardly through both diagrams are called "Q-curves"; we'll say more about them in a moment.

The orbit diagram of the middle picture will probably look familiar. At each value of c on the horizontal axis, it shows how the orbit of 0 behaves afterwards many iterations and in doing so, shows where f_c has alluring cycles and suggests where f_c behaves chaotically. This image has appeared in countless books and webpages, and indeed ``has go the most important icon of chaos theory'' ^PJS .

The bifurcation diagram in the left-hand picture may not exist so familiar. It shows the locations of all alluring and repelling cycles of f_c (upward to period 8 in this figure).

Clicking on either pic brings up a folio with information about both diagrams, [2004] and the bifurcation diagram in item, including the algorithm used to draw information technology. Pictures of bifurcation and orbit diagrams for other functions such as the logistic map and sin(cx) are included.

Now, delight be careful! The names I am using for these two diagrams are also used past authors such as Devaney, Strogatz, Peitgen/Jürgens/Saupe , only there are many books, manufactures, and webpages where the orbit diagram is chosen ``the bifurcation diagram'', and that which we call the ``bifurcation diagram'' is not discussed. 1 tin can likewise find places where bifurcation diagrams are discussed, but and so computer-drawn orbit diagrams are used in the illustrations without any mention of the difference(!). However, I accept been unable to find whatever other sources of detailed, computer-fatigued bifurcation diagrams. Our images bear witness that computers tin describe excellent pictures of both diagrams, and both are repleat with fascinating features.

One such characteristic shared by the ii diagrams is the illusion of shadowy polynomial curves - Q-curves - sweeping up and down as you look left-to-correct beyond either diagram. These curves are not implicitly fatigued in either diagram, and in fact appear as illusions in the ii diagrams for nearly contrary reasons. The Q-curves interact with the explicitly drawn curves of the bifurcation diagram in a beautiful fashion, and this interact is the field of study of an commodity we've called "The Shadow-Curves of the Orbit Diagram Permeate the Bifurcation Diagram, Too" (Download a PDF copy). This paper appeared in the September 2009 result of the International Journal of Bifurcation and Chaos.

Click hither for a quick, illustrated introduction to this interaction.

	It's Hither! The preliminary version 0.00 of my Windows plan OrbitsXP is set up to download; visit the download page. Please try information technology and report whatever problems and/or suggestions to me at `sross@bates.edu`. Endeavor It! [2005]
	Hither is an illustrated description [2005] of some of the capabilites of OrbitsXP, which is used to produce the newer pictures on this website.
	My coauthor Jody Sorensen and I were pleased to see our article "Will the Real Bifurcation Diagram Delight Stand!" appear as the encompass article in the Jan, 2000 result of The Higher Mathematics Journal. It is in this article in which the similarities and differences between the orbit diagram and the every bit beautiful bifurcation diagram are discussed, and the algorithm for drawing the bifurcation diagram is detailed. Download a PDF copy In August of 2001, Jody and I were honored to be called to receive the Mathematical Clan of America's George Pólya Accolade for this article! Visit the MAA website for more data about its awards. Click hither [2001] to visit my CMJ supplements page, for more information nearly and illustrations of the orbit and bifurcation diagrams, download my DOS programme "orbits.exe", and go html help files for "orbits".
	Visit this page [2001] for information and pictures about fractals. Julia sets, the Mandelbrot set, an blithe 3D rendering of the Lorenz Attractor, cauliflower, and Pipe Organ!? examples can all be found here. (Note: scroll downwards further for the Loftier Street Organ Page)
	Cheque out my fractal gallery for various images of the Mandelbrot prepare and some Julia sets, colored in various useful ways. [1997 only even so pertinent] ( Index )
	Since 1991 I have served as organist for High Street Congregational Church building, in Auburn, Maine. Our organ is a Skinner pipe organ, and it sounds wonderful in the reverberant sanctuary. Visit this page [2009] for a lilliputian history of the organ, the stop listing, and some interesting pictures, including the internal pipework, a visit to a much bigger organ, and more!
	Pictures of our family's pets, including Bailey the dog and the "destroyer" parakeets tin be found here. [2004]
	Some pictures of me with my students [2004] .
	Ane of my favorite hobbies is taking pictures of flowers. Here is a gallery of some of my pictures [2007] . I promise you bask them.

Unless otherwise noted, all photographs and diagrams throughout this web site were made by me, Chip Ross - with the full general exception of pictures in which I announced! All images by me © 2007. Please contact me if you would like to use them for whatever reason.

© 2011 by Chip Ross
Associate Professor of Mathematics
Bates Higher
Lewiston, ME 04240
email: sross@bates.edu (annotation: the outset letter of the alphabet is an "southward"!) sross sross sross sross NOT cross! NOT cross! NOT cross! For technical reasons across my command!