Sorry for the long post... When sciwise posted the Volterra Lotka (VL) formula here Nov. 7th, 2015 (his post is forwarded below) he later mentioned to me in a private email (that he likely intended to be posted to the [Fractint] list) that he inadvertently cut off the last few characters. He got the original formula from (and posted his BASIC code) at FreeBASIC.net on this page: http://www.freebasic.net/forum/viewtopic.php?f=8&t=24016 I'll repeat here the original source post from that page below. My previously posted VL images were made with the originally posted formula with the missing bailout test code. I've found that it's hypersensitive to maxiter value, which makes sense if you think about it. Using the code with the missing bailout test I made several fractals with the same parms, except for different maxiter value & confirmed this. I show a quick look at the corrected formula's images below. Notes on the original *freebasic.net* source post below: -------------------------------------------------------- - Look at the "18 Aug 93" date! - There are 2 formulas: "V-Euler" and "V-Heun" -- the one which is similar to the one sciwise previously posted with the last few characters (the bailout test) cut off. I put more about this below. - A set of parameters is provided for a "V-Heun" image. - "BOF" = "The Beauty of Fractals" ... "[is an award-winning] 1986 book by Heinz-Otto Peitgen and Peter Richter which publicises the fields of complex dynamics, chaos theory and the concept of fractals. It is lavishly illustrated and as a mathematics book became an unusual success." - https://en.wikipedia.org/wiki/The_Beauty_of_Fractals You can look a few pages of the book here: http://tinyurl.com/Beauty-of-Fractals-Images-of or: http://www.amazon.com/The-Beauty-Fractals-Complex-Dynamical/dp/3642617190 I'm hoping to locate my copy somewhere in my storage shed... Original FreeBASIC.net source post: ------------------------------------------------ =================================================== received from Ramiro Perez <RPEREZ@EARN.UTPVM1> 18 Aug 93 Path: unixg.ubc.ca!news.mic.ucla.edu!library.ucla.edu!europa.eng.gtefsd.com!darwin .sura.net!paladin.american.edu!auvm!VAXB.MIDDLESEX.AC.UK!DAVID1 Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU Newsgroups: bit.listserv.frac-l Via: uk.ac.mdx.vaxa; Mon, 27 Jun 1994 00:09:14 +0100 Message-ID: <FRAC-L%94062619112203@GITVM1.GATECH.EDU> Date: Mon, 27 Jun 1994 00:10:00 GMT Sender: "\"FRACTAL\" discussion list" <FRAC-L@GITVM1.BITNET> From: DAVID1@VAXB.MIDDLESEX.AC.UK Subject: Ramiro Perez Volterra .FRM Lines: 33 These are the Volterra-Lotka Formulas p 125 BOF received from Ramiro Perez: comment={received from Ramiro Perez <RPEREZ@EARN.UTPVM1> 18 Aug 93 } V-Euler{ x=real(pixel), y=imag(pixel), h=real(p1)/2: u=x-x*y, w=-y+x*y, c=x+h*(u+u), d=y+h*(w+w), x=c, y=d, z=x+flip(y), |z|<=p2 } V-Heun{ x=real(pixel), y=imag(pixel), p=real(p1), h=imag(p1)/2: u=x-x*y, w=-y+x*y, a=x+p*u, b=y+p*w, c=x+h*(u+(a-a*b)), d=y+h*(w+(-b+a*b)), x=c, y=d, z=x+flip(y), |z|<=p2 } Regards David Walter London england. VHeun { ; Volterra-Lottka reset=2004 type=formula formulafile=fractint.frm formulaname=V-Heun corners=0.000150015/6.00015/0/4.5 params=0.73899999999999999/0.73899999999999999/64/0 float=y maxiter=2048 inside=bof60 logmap=yes colors=@blues.map } ================================================== The above parm file "VHeun" (likely later than 1993) (with "frm:" added to the two formulas) creates: http://www.emarketingiseasy.com/TESTS/Fractals/VHeun00.gif which looks to be the built-in default Fractint VL fractal. NOTE: The original 1993 post above has this formula code: -------------------- V-Heun { . . . x=real(pixel), y=imag(pixel), p=real(p1), h=imag(p1)/2: -------------------- while sciwise's previously posted formula code has this code, which swaps the initialization of x & y with p & h 's initialization and vice versa: -------------------- V-HeunPH { . . . x=real(p1) y=imag(p1) p=real(pixel) h=imag(pixel): --------------------- The rest of the two formulas' (V-Heun/V-HeunPH) code is the same. Using: - Sciwise's corrected formula code (with his initialization mod above), - the same parm values as in the original 1993 post, - centered at (0, 0) with magnification 1.0, - with periodicity set to zero (to guarantee an accurate image -- at the expense of speed), produces this (anti-aliased) image: http://www.emarketingiseasy.com/TESTS/Fractals/VHeunS00.jpg (Click image to defeat browser resampling if needed.) And with a Paul Carlsonesque color map (taken from one of Albrect's parm files, I believe) the same image looks like this: http://www.emarketingiseasy.com/TESTS/Fractals/VHeunS01.jpg This is the par file and formula used for the above VHeunS00.jpg image: ---------------------------------------------------------------------------- -- VHeunS00 { ; sciwise's Volterra Lotka formula (with modified initialization) ; and bailout repair used. ; p, h variable. ; ; pars - taken from post with the 1993 V-Heun formula. ; frm - sciwise's modified ver of 1993 V-Heun formula: "V-HeunPH", repaired. ; reset=2004 type=formula formulafile=fractint.frm formulaname=V-HeunPH center-mag=0.0/0.0/1.0 params=0.73899999999999999/0.73899999999999999/64/0 float=y maxiter=2048 inside=bof60 logmap=yes colors=@blues.map } frm:V-HeunPH { ; Sciwise's FRM REPAIRED BY RESTORING BAILOUT TEST -HHL ; Also parameterized on initialization and bailout values. -HHL x=real(p1) y=imag(p1) p=real(pixel) h=imag(pixel): u=x-x*y w=-y+x*y a=x+p*u b=y+p*w c=x+h*(u+(a-a*b)) d=y+h*(w+(-b+a*b)) x=c y=d z=x+flip(y) |z|<=p2 } ---------------------------------------------------------------------------- ---------------------- Sorry for the over-long post. I think this is correct now. - Hal Lane ######################## # hallane@earthlink.net ######################## ======================================================= -----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of sciwise@ihug.co.nz Sent: Saturday, November 7, 2015 11:49 PM To: fractint@mailman.xmission.com Subject: [Fractint] Volterra Lotka PH This is the second part of the Volterra Lotka formulas , in this instance we are selecting an initial condition and varying the parameters p & h across the plane. There's now another FreeBASIC file to obtain the sequences from the map generated by V-HeunPH.frm . The coordinates I used were : Top left corner . -2.0 1.5 Bottom right corner . 2 -1.5 ---------------------------------------------------------------------------- ------- comment{Volterra Lotka , p , h variable . V-HeunPH.frm initial condition [ x = 0.8 , y=0.8 ], this initial condition might be determined from the V-Heun map . [ p , h ] varied over plane . x' = x - x*y y' = -y + x*y } V-HeunPH { x=0.8 y=0.8 p=real(pixel) h=imag(pixel): u=x-x*y w=-y+x*y a=x+p*u b=y+p*w c=x+h*(u+(a-a*b)) d=y+h*(w+(-b+a*b)) x=c y=d z=x+flip(y) |z| _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint --- --- This email has been checked for viruses by Avast antivirus software. https://www.avast.com/antivirus