Dear H.Lane & fractint group , Thank you for being so precise , this does indeed correct the formula , in particular the bailout test. Since this post I've been using the bof61 option for the inner iterations. Also I'm only looking at the positive p and h axis as these are the most relevant for this type of iteration ; again I'm using the blues.map for the colorization. If you look at the Volterra Lotka fractal in the Beauty of Fractals you may find that the axis are incorrectly labeled p and h , whereas they should be x and y . This is how the Volterra Lotka fractal is generated within fractint , by setting p and h to constants and varying x and y. As noted my variation on the theme sets x and y to a constant and varies p and h. My exploration of this fractal is something that I'm only doing intermittently. Hal is better versed in using fractint than I am . I'm using xfractint 20.04.10 , to load a *.par file I issue these commands ; @ , F5 . As mentioned in a previous post , there is a wealth of formula and parameter files for fractint , the Volterra Lotka just being one of a family of numerical integrators. At this stage I don't know how generally applicable my approach is to a broader range of numerical integrator. It almost feels like heresy to use fractint to delve into these types of iteration ; as B.Mandelbrot noted fractal curves don't tend to have derivatives and aren't amenable to the classical analytical methods. sciwise On 20/11/2015 04:55, Hal Lane wrote:

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 [1]

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 [2]

You can look a few pages of the book here:

http://tinyurl.com/Beauty-of-Fractals-Images-of [3]

or:

http://www.amazon.com/The-Beauty-Fractals-Complex-Dynamical/dp/3642617190 [4]

I'm hoping to locate my copy somewhere in my storage shed...

Original FreeBASIC.net source post:

------------------------------------------------

===================================================

received from Ramiro Perez 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 [6]

Newsgroups: bit.listserv.frac-l Via: uk.ac.mdx.vaxa; Mon, 27 Jun 1994 00:09:14 +0100 Message-ID: Date: Mon, 27 Jun 1994 00:10:00 GMT Sender: ""FRACTAL" discussion list" From: DAVID1@VAXB.MIDDLESEX.AC.UK [9]

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 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|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 [11]

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),

- centered at (0,

- the same parm values as in the original 1993 post, 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 [12]

(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 [13]

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 [14]

}

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|hallane@earthlink.net

########################

=======================================================

-----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com [15]] On Behalf Of

sciwise@ihug.co.nz [16]

Sent: Saturday, November 7, 2015 11:49 PM

To: fractint@mailman.xmission.com [17]

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| _______________________________________________

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Links: ------ [1] http://www.freebasic.net/forum/viewtopic.php?f=8|+|amp|+|t=24016 [2] https://en.wikipedia.org/wiki/The_Beauty_of_Fractals [3] http://tinyurl.com/Beauty-of-Fractals-Images-of [4] http://www.amazon.com/The-Beauty-Fractals-Complex-Dynamical/dp/3642617190 [5] mailto:RPEREZ@EARN.UTPVM1 [6] mailto:NETNEWS@AUVM.AMERICAN.EDU [7] mailto:FRAC-L%94062619112203@GITVM1.GATECH.EDU [8] mailto:FRAC-L@GITVM1.BITNET [9] mailto:DAVID1@VAXB.MIDDLESEX.AC.UK [10] mailto:RPEREZ@EARN.UTPVM1 [11] http://www.emarketingiseasy.com/TESTS/Fractals/VHeun00.gif [12] http://www.emarketingiseasy.com/TESTS/Fractals/VHeunS00.jpg [13] http://www.emarketingiseasy.com/TESTS/Fractals/VHeunS01.jpg [14] mailto:colors=@blues.map [15] mailto:fractint-bounces@mailman.xmission.com [16] mailto:sciwise@ihug.co.nz [17] mailto:fractint@mailman.xmission.com [18] mailto:Fractint@mailman.xmission.com [19] https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint [20] https://www.avast.com/antivirus