I found and downloaded the text you mentioned this appears to be a new development. Some decades ago I determined the first derivative of the Mandelbrot series , only within the last few weeks , using Maxima CAS , have I generalized this to other fractals and the second derivative thereof. I've done this so that the community might construct other interesting fractals whilst using a bit more math. Sciwise On 06/12/2016 08:35, Jovan Trujillo wrote:
Well I now have new motivation to read Differential Equations on Fractals. Thanks sciwise! I'll have to test out this code next week when finals are over and I finally graduate.
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On Dec 5, 2016, at 12:00 PM, fractint-request@mailman.xmission.com [1] wrote: Send Fractint mailing list submissions to fractint@mailman.xmission.com [2] To subscribe or unsubscribe via the World Wide Web, visit https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint [3] or, via email, send a message with subject or body 'help' to fractint-request@mailman.xmission.com [4] You can reach the person managing the list at fractint-owner@mailman.xmission.com [5] When replying, please edit your Subject line so it is more specific than "Re: Contents of Fractint digest..." Today's Topics: 1. Maxima cas for Fractint (sciwise@ihug.co.nz [6])
Message: 1 Date: Mon, 05 Dec 2016 18:22:18 +1300 From: sciwise@ihug.co.nz [7] To: Fractint and General Fractals Discussion Subject: [Fractint] Maxima cas for Fractint Message-ID: Content-Type: text/plain; charset=UTF-8 This is code that I've written in Maxima CAS . This finds the first and second derivative of an iterated function , g(z,c) . To construct a fractal substitute z:g(z,c) with the equation for g(z,c) , rather than calling a function. Similarly for z1:g1(z,z1,c) and z2:g2(z,z1,z2,c) . I've provided an example of what this should look like. The validity of this approach is tested by symbolically calculating the second derivative at each iteration and comparing this with the second derivative from the above method. This is done for three iterations and , if correct , will display TRUE. The file difffrac1h.mac /* . (c) 2016 , sciwise@ihug.co.nz [10] . A general method for defining the first and second derivatives of an iterated fractal function , g(z,c) . Some care is required to ensure that g(0,c) is defined . . */ /* Example Fractint code . d2zmand(XAXIS){; Edward Montague ; Mandelbrot series = z ; 1st Derivative of Mandelbrot series = z1. ; 2nd Derivative of Mandelbrot series = z2. z = Pixel z1 = 1 z2 = 0: z2 = 2*z1*z1 +2*z2*z z1 = 2*z*z1 + 1 z = z*z+Pixel |z2| z1 */ eq:diff(g(z,c),z,1)*z1 + diff(g(0,c),c); define(g1(z,z1,c),eq)$ /* Second Derivative --> z2 */ eq:diff(g(z,c),z,2)*z1*z1 + diff(g(z,c),z)*z2 + diff(g(0,c),c,2); define(g2(z,z1,z2,c),eq)$ /* Initial values */ z : c; z1 : diff(z,c); z2 : diff(z,c,2); /* First iteration. */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c); eq:diff(z,c,2); is(equal(eq,z2)); /* Second iteration. */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c); eq:diff(z,c,2)$ is(equal(eq,z2)); /* Third iteration. */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c)$ eq:diff(z,c,2)$ is(equal(eq,z2)); /* */ And the file difffrac1h.wxm /* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/ /* [ Created with wxMaxima version 11.08.0 ] */ /* [wxMaxima: comment start ] . (c) 2016 , sciwise@ihug.co.nz [11] . A general method for defining the first and second derivatives of an iterated fractal function , g(z,c) . Some care is required to ensure that g(0,c) is defined . . [wxMaxima: comment end ] */ /* [wxMaxima: comment start ] Example Fractint code . d2zmand(XAXIS){; Edward Montague ; Mandelbrot series = z ; 1st Derivative of Mandelbrot series = z1. ; 2nd Derivative of Mandelbrot series = z2. z = Pixel z1 = 1 z2 = 0: z2 = 2*z1*z1 +2*z2*z z1 = 2*z*z1 + 1 z = z*z+Pixel |z2| z1 [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ eq:diff(g(z,c),z,1)*z1 + diff(g(0,c),c); define(g1(z,z1,c),eq)$ /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] Second Derivative --> z2 [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ eq:diff(g(z,c),z,2)*z1*z1 + diff(g(z,c),z)*z2 + diff(g(0,c),c,2); define(g2(z,z1,z2,c),eq)$ /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] Initial values [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ z : c; z1 : diff(z,c); z2 : diff(z,c,2); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] First iteration. [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c); eq:diff(z,c,2); is(equal(eq,z2)); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] Second iteration. [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c); eq:diff(z,c,2)$ is(equal(eq,z2)); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] Third iteration. [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c)$ eq:diff(z,c,2)$ is(equal(eq,z2)); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* Maxima can't load/batch files which end with a comment! */ "Created with wxMaxima"$ ------------------------------ Subject: Digest Footer _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com [12] https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint [13] ------------------------------ End of Fractint Digest, Vol 166, Issue 1 ****************************************
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