While looking through a used book I bought (Fractals Images of Chaos by Hans Lauwerier) I saw a formula I wasn't familiar with. The formula was discovered in 1969 by French mathematician M. Henon. I entered it into Fracton, played with it a bit, and found a few fractals I liked. Henon is listed in FractInt's built in formula list but I have never seen anyone use it. I have no way to tell if FractInt's Henon is the same as this one. Here is a page with an image: http://www.fracton.org/fmlposts/henon_jko.html Here is the par file used to generate the fractal: Henon_JKO { ; Exported from Fracton. reset=2004 type=formula formulafile=fracton.frm formulaname=F_20101115_0906 passes=1 float=y center-mag=0.32/0.15/0.53333334/1/0/0 params=-0.0235/0.488/4.6/0/0/0/0/0/0/0 maxiter=190 proximity=0 colors=000aaateOthNtjNtlMtnMtoLtqLtsKtuKswJswJsvJs\ uJsuJstJssJssJsrKsqKspKspKsoKsnKsnKsmKslLslLskLsjL\ siLsiLshLsgLsgMsfMseMseMsdMscMsbMsbMsaNs`Ns`Ns_NsZ\ NsZNsYNsXNsWOsWOsVOsUOsUOsTOsSOsSOsRPsQPsPPsPPsOPs\ NPsNPsMPsLQsLQsKQsJQsIQsIQsHQsGQsGRsFRsERsERsDRsCR\ sBRsBRsASs9Ss9Ss8Ss7Ss7Ss6Ss5Sr4Tq4Tp4To5Tn5Sl6Sk6\ Sj7Si7Rh8Rf8Re9Rd9QcAQbAQ`BQ_BPZCPYCPWDOWDOWDOWDOW\ DOWDOVDOVDOVDOVDOVDOUDOUDOUDOUDOUDOTDOTDOTDOTDOTDO\ TDOSDOSDOSDOSDOSDORCORCORCORCORCOQCOQCOQCOQCOQCOQC\ OPCOPCOPCPPCPPCPOCPOCPOCPOCPOCPNCPNCPNCPNCPNCPNCPM\ BPMBPMBPMBPMBPLBPLBPLBPLBPLBPKBPKBPKBPKBPKBPKBPJBP\ JBPJBPJBPJBPIBPIBPIBPIBPIBPHAQIBQJDQLEQMGQOHQPJQRK\ QSMQUNQVPQWQQYSQZTQ`VRaWRcYRdZRf`RgaRhcRjdRkfRmgRn\ iRpjRqlRsnSrnSpnSnnSmnSknSinSgnSfnTdnTbnTanT_nTYnT\ WnTVnTTnURnUQnUOnUMnUKnUJnUHnUFnVEnVCnVAnV8nV7nV5n\ V3nV1oW1oW1oW1oW1oW1pV1pV } frm:F_20101115_0906 { ; Henon's Expression ; From Fractals Images Of Chaos by Hans Lauwerier p126 ; x' = y ; y' = -x +2ay + by^2 ; Time = 0.4 sec at 640 x 480 with no anti-aliasing ; using Fracton on Mac Pro 2 x 2.8 GHz Quad-Core Xeon x=real(pixel),y=imag(pixel), a=2*real(p1),b=imag(p1),; p1 - Julia constant bailout=real(p2),; real p2 - bailout @beginloop, temp=y, y=a*y+b*y*y-x, x=temp, z=x+flip(y), |z|<=bailout } -- Mike Frazier www.fracton.org
In article <AANLkTinv7iezk+8s07XtG_wG2D59QMT_8iC2rC4eB1pz@mail.gmail.com>, Mike Frazier <fractonorg@gmail.com> writes:
; Henon's Expression ; From Fractals Images Of Chaos by Hans Lauwerier p126 ; x' = y ; y' = -x +2ay + by^2
Fractint uses the following formula for the henon type: Orbit in two dimensions defined by: x(n+1) = 1 + y(n) - a*x(n)*x(n) y(n+1) = b*x(n) Two parameters: a and b You can find this in the fractint docs. Lots of these mathematicians (Henon, Poincare, etc.) didn't study a single system, they studied families of systems and in many cases multiple families. Its very often the case that any system with non-linear terms (i.e. y^2 or x^2) can exhibit chaotic behavior for some particular set of parameters. -- "The Direct3D Graphics Pipeline" -- DirectX 9 draft available for download <http://legalizeadulthood.wordpress.com/the-direct3d-graphics-pipeline/> Legalize Adulthood! <http://legalizeadulthood.wordpress.com>
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