Lyapgreen.bmp 289 KB
Lyapgreen.jpeg 36 KB
Some notes about the following pictures: they are based on the Lyapunov exponent of iterated systems with periodically changing parameter. For more mathematical details, see for example Mario Markus, Scientific American 5/1995.
The iterated function is f(x) = b sin2(x+r), with r as the alternating parameter. r is changing between values A and B, and the pictures below show the AB-plane. The starting value of x is in any case = 0 (but the starting value doesn't matter much). Since f is a periodic function and r describes a function shift, the AB-plane itself is periodic. The coloration for a given value of A and B is based on the Lyapunov exponent of the function f and starting value x. This is computed as the sum of the logarithm of the first derivation of f for the first thousend-or-something iterations of fn(x), that means:
lambda = Sum(log(f'(fn(x)))/N; m < n < N)
The first 50 or 500 or 1000 or m iterations are not taken into consideration, just the iterations from m to N. The most interesting thing about lambda is wether it is greater or smaller than 0: the case <0 means that small changes in the starting x tends to vanish, while the case >0 means that small changes in x tends to grow. The following pictures show AB-planes with lambda values represented as different colors. In most pictures, r changes between A and B in the manner ABABABAB..., but on some pictures, more complicated patterns are choosen. In most cases,
Since the pattern on the AB-plane is periodic, these pictures form natural tilings, which could be used as desktop tilings; that for, I added not only a JPEG version, but also a BMP version of every file; also, I included a web page with the according picture as a background tiling. Click on one of the pictures below to see a sample tiling in a new window.
The following contains a bug, but the images are nice anyway, so I didn’t delete this page.
If you want to reproduce the pictures below, you need a fractal programm like Ultra Fractal and some additional files I have written and stored in ultralyap.zip (5 KB, 23 KB unzipped, contains jan.ucl, jan.ufm, jan.ugr and jan.upr, files that need to be placed in the according Ultra Fractal folders).
There are still infinite many new pictures hidden in this formula.
Please note that the pictures on this page are based on a somewhat defective programm; see Lyapunov Tilings Part 2 for details and also some new pictures.
I also added Lyapunov Tilings Part 3 with 30 new tiles and 26 zooms, making a total of 56 additional pictures.
Lyapgreen.bmp 289 KB
Lyapgreen.jpeg 36 KB
Lyapgreen2.bmp 289 KB
Lyapgreen2.jpeg 32 KB
Lyapgreen3.bmp 289 KB
Lyapgreen3.jpeg 37 KB
Lyapgreen4.bmp 289 KB
Lyapgreen4.jpeg 20 KB
Lyapgreen5.bmp 289 KB
Lyapgreen5.jpeg 31 KB
Lyapgreen6.bmp 289 KB
Lyapgreen6.jpeg 25 KB
GreenOrange.bmp 289 KB
GreenOrange.jpeg 33 KB
Lyapblue.bmp 289 KB
Lyapblue.jpeg 46 KB
Lyapblue2.bmp 289 KB
Lyapblue2.jpeg 36 KB
Lyapblue3.bmp 289 KB
Lyapblue3.jpeg 29 KB
Lyapblue4.bmp 289 KB
Lyapblue4.jpeg 33 KB
Aleksandr Mikhailovich Lyapunov was born 6 June 1857 in Yaroslavl (Russia) and died 3 November 1918 in Odessa; he committed suicide three days after the death of his wive.
Main Lyapunov Index Page
Main Jan Page