I tried Sciwise's formula derived from a differential equation (his 2 posts are in my PAR file below.) I also pasted the PAR file I used at the bottom of this email. I used the color map: default.map since its strongly contrasting adjacent colors can help in understanding a new fractal. His original formula with bailout: |z|<1 gives these images: http://www.emarketingiseasy.com/Sciwise/S161011B.jpg http://www.emarketingiseasy.com/Sciwise/S161011C.gif http://www.emarketingiseasy.com/Sciwise/S161011D.gif The "B" image shows the color map. The fractal never uses colors (iterations) other than: 1, 2, and 3. Zooming way out gives an all blue image. In his 2nd post re this differential equation, Sciwise says: ---------------------------------------- If you look at the fractal generated by my differential equation, for |z| > 10000 ; [I assume Sciwise meant to type "<". hhl] then the traditional Mandelbrot set appears plus embellishments. ------------------------------------------- I made his original formula's bailout value a parameter: real(P1) and got these images using 10,000 for real(P1) : http://www.emarketingiseasy.com/Sciwise/S161011F.gif Details in "F": http://www.emarketingiseasy.com/Sciwise/S161011G.gif http://www.emarketingiseasy.com/Sciwise/S161011H.gif I then tried adjusting the bailout value: real(P1) a bit: |z| > 50: http://www.emarketingiseasy.com/Sciwise/S161011J.gif |z| > 100: http://www.emarketingiseasy.com/Sciwise/S161011K.gif |z| > 500: http://www.emarketingiseasy.com/Sciwise/S161011L.gif |z| > 10,000: http://www.emarketingiseasy.com/Sciwise/S161011M.gif which is a zoom into "F" above. |z| > 1,000,000 and zooming in some adds details: http://www.emarketingiseasy.com/Sciwise/S161011N.gif A couple of further zooms into "N" are here: http://www.emarketingiseasy.com/Sciwise/S161011P.gif http://www.emarketingiseasy.com/Sciwise/S161011Q.gif These images aren't visual "eye candy," but remember -- this is research... I tried to make the ending letter of my PAR names match the ending letter of the matching image file but I didn't succeed. My 3 PARs differ in real(P1). P1 can also be set in Fractint's "Z" screen. Thanks, Sciwise! - Hal Lane ######################## # hallane@earthlink.net ######################## --------- Begin Hal's PAR file: ----------------- comment{ Author: sciwise Date: 2016-10-11 05:55 -400 To: Fractint and General Fractals Discussion Subject: [Fractint] A Differential Equation [D.E.] Continuing from previous posts on the topic of the derivative[s] of the Mandelbrot Series. I now attempt to examine the regions of stability for this D.E. y'' + y' + y'*y' + y - exp(y*y') = 0 Where I set y equal to the Mandelbrot series, with initial conditions determined by the value of Pixel. I remain uncertain at this stage as to how to interpret the results. I'm assuming that the regions with the fewest iterations are what we require. Another approach to solving second order differential equations is to use the Euler - Heun method; this however provides less information as it's usually implemented as a purely numerical approach. Non linear differential equations have been difficult to solve as the elementary functions don't provide the scope required. Anyway here's the code. de2bmand(XAXIS) [ <---<<!!!; Edward Montague ; 2nd order non linear differential equation = z. ; y'' + y' + y'*y' + y - exp(y*y') = 0 ; 2nd derivative of mandelbrot series = w. ; 1st derivative of mandelbrot series = u. ; Mandelbrot series = v v = Pixel u = 1 w = 0 z = 0: w = 2*u*u+2*w*v u = 2*v*u + 1 z = w +u + u*u + v - exp(v*u) v = v*v+Pixel |z|<1 ] <---<<!!! Sciwise also posted: If you look at the fractal generated by my differential equation, for |z| > 10000 ; [I assume Sciwise meant to type "<". hhl] then the traditional Mandelbrot set appears plus embellishments. I shall examine a more well known differential equation at some other stage. There are various to choose from. ---------- END OF COMMENT ---------- } de2bmand_161011_B { ; Differential Eqn. of Mbrot Series. ; Edward Montague ; Parent fractal ; Color map = default.map ; ; Richard's Fractint for Windows ; Ver 20.99.8 reset=2099 type=formula formulafile=161011d2.PAR formulaname=de2bmand passes=1 params=1/0/0/0/0/0/0/0/0/0 center-mag=-0.620156/2.22045e-016/0.5186722 float=y maxiter=10000 inside=0 colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\ zLzzz000555<3>HHHKKKOOO<3>ccchhhmmmssszzz00z<3>z0z\ <3>z00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zVz<3>zVV<3>zz\ V<3>VzV<3>Vzz<2>Vbzhhz<3>zhz<3>zhh<3>zzh<3>hzh<3>h\ zz<2>hlz00S<3>S0S<3>S00<3>SS0<3>0S0<3>0SS<2>07SEES\ <3>SES<3>SEE<3>SSE<3>ESE<3>ESS<2>EHSKKS<2>QKSSKSSK\ QSKOSKMSKK<2>SQKSSKQSKOSKMSKKSK<2>KSQKSSKQSKOSKMS0\ 0G<3>G0G<3>G00<3>GG0<3>0G0<3>0GG<2>04G88G<2>E8GG8G\ G8EG8CG8AG88<2>GE8GG8EG8CG8AG88G8<2>8GE8GG8EG8CG8A\ GBBG<2>FBGGBGGBFGBDGBCGBB<2>GFBGGBFGBDGBCGBBGB<2>B\ GFBGGBFGBDGBCG000<6>000 } de2bmand_161011_F { ; Differential Eqn. of Mbrot Series. ; Edward Montague ; Parent fractal ; Color map = default.map ; ; Richard's Fractint for Windows ; Ver 20.99.8 reset=2099 type=formula formulafile=161011d2.PAR formulaname=de2bmand passes=1 params=10000/0/0/0/0/0/0/0/0/0 center-mag=-0.620156/2.22045e-016/0.5186722 float=y maxiter=10000 inside=0 colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\ zLzzz000555<3>HHHKKKOOO<3>ccchhhmmmssszzz00z<3>z0z\ <3>z00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zVz<3>zVV<3>zz\ V<3>VzV<3>Vzz<2>Vbzhhz<3>zhz<3>zhh<3>zzh<3>hzh<3>h\ zz<2>hlz00S<3>S0S<3>S00<3>SS0<3>0S0<3>0SS<2>07SEES\ <3>SES<3>SEE<3>SSE<3>ESE<3>ESS<2>EHSKKS<2>QKSSKSSK\ QSKOSKMSKK<2>SQKSSKQSKOSKMSKKSK<2>KSQKSSKQSKOSKMS0\ 0G<3>G0G<3>G00<3>GG0<3>0G0<3>0GG<2>04G88G<2>E8GG8G\ G8EG8CG8AG88<2>GE8GG8EG8CG8AG88G8<2>8GE8GG8EG8CG8A\ GBBG<2>FBGGBGGBFGBDGBCGBB<2>GFBGGBFGBDGBCGBBGB<2>B\ GFBGGBFGBDGBCG000<6>000 } de2bmand_161011_J { ; Differential Eqn. of Mbrot Series. ; Edward Montague ; Parent fractal ; Color map = default.map ; ; Richard's Fractint for Windows ; Ver 20.99.8 reset=2099 type=formula formulafile=161011d2.PAR formulaname=de2bmand passes=1 params=50/0/0/0/0/0/0/0/0/0 center-mag=-0.620156/2.22045e-016/0.5186722 float=y maxiter=10000 inside=0 colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\ zLzzz000555<3>HHHKKKOOO<3>ccchhhmmmssszzz00z<3>z0z\ <3>z00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zVz<3>zVV<3>zz\ V<3>VzV<3>Vzz<2>Vbzhhz<3>zhz<3>zhh<3>zzh<3>hzh<3>h\ zz<2>hlz00S<3>S0S<3>S00<3>SS0<3>0S0<3>0SS<2>07SEES\ <3>SES<3>SEE<3>SSE<3>ESE<3>ESS<2>EHSKKS<2>QKSSKSSK\ QSKOSKMSKK<2>SQKSSKQSKOSKMSKKSK<2>KSQKSSKQSKOSKMS0\ 0G<3>G0G<3>G00<3>GG0<3>0G0<3>0GG<2>04G88G<2>E8GG8G\ G8EG8CG8AG88<2>GE8GG8EG8CG8AG88G8<2>8GE8GG8EG8CG8A\ GBBG<2>FBGGBGGBFGBDGBCGBB<2>GFBGGBFGBDGBCGBBGB<2>B\ GFBGGBFGBDGBCG000<6>000 } frm:de2bmand(XAXIS){; Edward Montague ; ; 2nd order non linear differential equation: ; ; z = y'' + y' + y'*y' + y - exp(y*y') = 0 ; ; v = Mandelbrot series [z*z+c -- HHL] ; u = 1st derivative of mandelbrot series ; w = 2nd derivative of mandelbrot series ; v = Pixel u = 1 w = 0 z = 0: w = 2*u*u+2*w*v u = 2*v*u+1 z = w +u + u*u + v - exp(v*u) v = v*v+Pixel |z|<P1 } --------- End of Hal's PAR file ----------------- --- This email has been checked for viruses by Avast antivirus software. https://www.avast.com/antivirus